https://wiki.swarma.org/api.php?action=feedcontributions&feedformat=atom&user=Typhoid集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织 - 用户贡献 [zh-cn]2026-11-03T03:09:09Z用户贡献MediaWiki 1.35.0https://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19784拐点2020-12-07T03:58:07Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，若曲线是可导性类的函数图像，那么在曲线拐点处二阶导数为0，并且改变了它的符号(从正到负或从负到正)。二阶导数为0但其符号不变的点有时称为波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
定义<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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在微分几何中，拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，当其仅当一阶导数在x处具有孤立极值点时（这不同于极值点的说法），可微函数图才在(x, f(x))处拥有拐点。也就是说，在某些邻域中，该点是唯一具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上切线与曲线相交的点。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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下降拐点的两边导数都是负数，即在该点附近函数减小。上升拐点的两边导数都为正，即在该点附近函数增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，当且仅当切线与曲线(在切点处)的交点数大于2时，非奇点为拐点。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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其主要结果是代数曲线拐点的集合与曲线同海森曲线的交点集合一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，若某点处曲率从正变为负或从负变为正，即改变曲率符号，则该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条二次可微函数的光滑曲线，拐点处上的二次导数为0并且改变曲率符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此相反。切线是蓝色的，该处曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，并且红色的是拐点: 0,/2和]。<br />
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==A necessary but not sufficient condition==<br />
必要非充分条件<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}在x0处存在，并且x0是该函数的拐点，那么(x < sub > 0 </sub >)0} ，那么即使存在任意阶的导数，这个条件对于有拐点也是不充分的。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。若最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点被统称为拐点。对于给定的 x < sup > 4 </sup > }的函数，波动点是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在该处有一个拐点。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件则要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近具有相反的符号。<br />
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==Categorization of points of inflection==<br />
拐点的分类<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 -x }在点(0,0)处二阶导数为0，但它不是拐点，因为其四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是否为0来进行分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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驻点并不是局部极值点。普遍地，在多实变量函数的前提下，不是局部极值点的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点的例子是在x<sup>3 图上的点（0,0），其切线是x轴。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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一个非驻点的例子是x < sup > 3 + ax }图上的（0,0），对于任意非零的a，在原点处的切线是ax }}}}<br />
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==Functions with discontinuities==<br />
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非连续性函数<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]临界点<br />
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* [[Ecological threshold]]生态阈值<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]海塞配置 被椭圆曲线上九个拐点所组成<br />
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* [[Ogee]], an architectural form with an inflection point S形曲线，具有一个拐点的建筑型式曲线<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature顶点，曲线的局部最小或局部最大值点<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19783拐点2020-12-07T03:34:57Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
定义<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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在微分几何中，拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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下降拐点的两边导数都是负数，即在该点附近函数减小。上升拐点的两边导数都为正，即在该点附近函数增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，当且仅当切线与曲线(在切点处)的交点数大于2时，非奇点为拐点。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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其主要结果是代数曲线拐点的集合与曲线同海森曲线的交点集合一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，若某点处曲率从正变为负或从负变为正，即改变曲率符号，则该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条二次可微函数的光滑曲线，拐点处上的二次导数为0并且改变曲率符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
必要非充分条件<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}在x0处存在，并且x0是该函数的拐点，那么(x < sub > 0 </sub >)0} ，那么即使存在任意阶的导数，这个条件对于有拐点也是不充分的。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。若最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点被统称为拐点。对于给定的 x < sup > 4 </sup > }的函数，波动点是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在该处有一个拐点。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件则要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近具有相反的符号。<br />
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==Categorization of points of inflection==<br />
拐点的分类<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 -x }在点(0,0)处二阶导数为0，但它不是拐点，因为其四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是否为0来进行分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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驻点并不是局部极值点。普遍地，在多实变量函数的前提下，不是局部极值点的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点的例子是在x<sup>3图上的点（0,0），其切线是x轴。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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一个非驻点的例子是x < sup > 3 + ax }图上的（0,0），对于任意非零的a，在原点处的切线是ax }}}}<br />
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==Functions with discontinuities==<br />
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非连续性函数<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]临界点<br />
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* [[Ecological threshold]]生态阈值<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]海塞配置 被椭圆曲线上九个拐点所组成<br />
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* [[Ogee]], an architectural form with an inflection point S形曲线，具有一个拐点的建筑型式曲线<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature顶点，曲线的局部最小或局部最大值点<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19782拐点2020-12-07T03:15:25Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
定义<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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在微分几何中，拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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下降拐点的两边导数都是负数，即在该点附近函数减小。上升拐点的两边导数都为正，即在该点附近函数增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，当且仅当切线与曲线(在切点处)的交点数大于2时，非奇点为拐点。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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其主要结果是代数曲线拐点的集合与曲线同海森曲线的交点集合一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，若某点处曲率从正变为负或从负变为正，即改变曲率符号，则该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条二次可微函数的光滑曲线，拐点处上的二次导数为0并且改变曲率符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
必要非充分条件<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
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==Categorization of points of inflection==<br />
拐点的分类<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 -x }在点(0,0)处二阶导数为0，但它不是拐点，因为其四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是否为0来进行分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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驻点并不是局部极值点。普遍地，在多实变量函数的前提下，不是局部极值点的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点的例子是在x<sup>3图上的点（0,0），其切线是x轴。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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一个非驻点的例子是x < sup > 3 + ax }图上的（0,0），对于任意非零的a，在原点处的切线是ax }}}}<br />
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==Functions with discontinuities==<br />
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非连续性函数<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]<br />
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* [[Ecological threshold]]<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
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* [[Ogee]], an architectural form with an inflection point<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19781拐点2020-12-07T02:54:57Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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微分几何中的拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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拐点的下降点是一个拐点，在这个点的两边导数都是负数; 换句话说，它是一个拐点，在这个点附近函数正在减少。拐点的上升点是一个导数在点的两边都是正的点; 换句话说，它是一个拐点，在这个点附近函数正在增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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主要结果是代数曲线拐点的集合与曲线与黑森曲线的交集相一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，如果有符号的曲率从正变为负或从负变为正，即改变符号，那么该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条光滑曲线，它是一个二次可微函数的图形，一个二次拐点是图形上的一个点，在这个点上二次导数有一个孤立的零并且改变符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
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==Categorization of points of inflection==<br />
==拐点的分类==<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 -x }在点(0,0)处二阶导数为0，但它不是拐点，因为其四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是否为0来进行分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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驻点并不是局部极值点。普遍地，在多实变量函数的前提下，不是局部极值点的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点的例子是在x<sup>3图上的点（0,0），其切线是x轴。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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一个非驻点的例子是x < sup > 3 + ax }图上的（0,0），对于任意非零的a，在原点处的切线是ax }}}}<br />
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==Functions with discontinuities==<br />
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==非连续性函数==<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]<br />
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* [[Ecological threshold]]<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
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* [[Ogee]], an architectural form with an inflection point<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19780拐点2020-12-07T02:37:08Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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微分几何中的拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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拐点的下降点是一个拐点，在这个点的两边导数都是负数; 换句话说，它是一个拐点，在这个点附近函数正在减少。拐点的上升点是一个导数在点的两边都是正的点; 换句话说，它是一个拐点，在这个点附近函数正在增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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主要结果是代数曲线拐点的集合与曲线与黑森曲线的交集相一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，如果有符号的曲率从正变为负或从负变为正，即改变符号，那么该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条光滑曲线，它是一个二次可微函数的图形，一个二次拐点是图形上的一个点，在这个点上二次导数有一个孤立的零并且改变符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
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==Categorization of points of inflection==<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 </sup >-x }在点(0,0)处有一个零的二阶导数，但它不是拐点，因为四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是零还是非零来分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点拐点的例子是 x < sup > 3 </sup > }图上的点。切线是-轴，它在此处切割图形。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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非平稳拐点的一个例子是 x < sup > 3 </sup > + ax }图上对任意非零点的拐点。原点处的切线是直线 ax }}}} ，它在此处切割图形。<br />
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==Functions with discontinuities==非连续性函数<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]<br />
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* [[Ecological threshold]]<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
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* [[Ogee]], an architectural form with an inflection point<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19765拐点2020-12-06T14:58:16Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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微分几何中的拐点是改变曲率符号的点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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拐点的下降点是一个拐点，在这个点的两边导数都是负数; 换句话说，它是一个拐点，在这个点附近函数正在减少。拐点的上升点是一个导数在点的两边都是正的点; 换句话说，它是一个拐点，在这个点附近函数正在增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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主要结果是代数曲线拐点的集合与曲线与黑森曲线的交集相一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，如果有符号的曲率从正变为负或从负变为正，即改变符号，那么该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条光滑曲线，它是一个二次可微函数的图形，一个二次拐点是图形上的一个点，在这个点上二次导数有一个孤立的零并且改变符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
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==Categorization of points of inflection==<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 </sup >-x }在点(0,0)处有一个零的二阶导数，但它不是拐点，因为四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是零还是非零来分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点拐点的例子是 x < sup > 3 </sup > }图上的点。切线是-轴，它在此处切割图形。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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非平稳拐点的一个例子是 x < sup > 3 </sup > + ax }图上对任意非零点的拐点。原点处的切线是直线 ax }}}} ，它在此处切割图形。<br />
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==Functions with discontinuities==<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下改变凹度。相反，它们可以改变垂直渐近线或不连续性周围的凹度。例如，函数 < math > x mapsto frac1x </math > 是凹的表示负的，凸的表示正的，但是它没有拐点，因为0不在函数的域中。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]<br />
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* [[Ecological threshold]]<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
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* [[Ogee]], an architectural form with an inflection point<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19680拐点2020-12-05T10:50:19Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何中，拐点是光滑平面曲线上的改变曲率符号的一个点。特别地，在函数图像中，拐点处函数从凹(向下)变为凸(向上) ，反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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微分几何中的拐点是曲率改变其符号的曲线的拐点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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拐点的下降点是一个拐点，在这个点的两边导数都是负数; 换句话说，它是一个拐点，在这个点附近函数正在减少。拐点的上升点是一个导数在点的两边都是正的点; 换句话说，它是一个拐点，在这个点附近函数正在增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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主要结果是代数曲线拐点的集合与曲线与黑森曲线的交集相一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，如果有符号的曲率从正变为负或从负变为正，即改变符号，那么该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条光滑曲线，它是一个二次可微函数的图形，一个二次拐点是图形上的一个点，在这个点上二次导数有一个孤立的零并且改变符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
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如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
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2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
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2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
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2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
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==Categorization of points of inflection==<br />
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[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
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x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
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X < sup > 4 </sup >-x }在点(0,0)处有一个零的二阶导数，但它不是拐点，因为四阶导数是一阶非零导数(三阶导数也是零)。<br />
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Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
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Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
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拐点也可以根据(x)}是零还是非零来分类。<br />
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* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
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* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
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A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
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A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
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一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。<br />
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An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
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An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
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一个驻点拐点的例子是 x < sup > 3 </sup > }图上的点。切线是-轴，它在此处切割图形。<br />
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An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
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An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
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非平稳拐点的一个例子是 x < sup > 3 </sup > + ax }图上对任意非零点的拐点。原点处的切线是直线 ax }}}} ，它在此处切割图形。<br />
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==Functions with discontinuities==<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
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Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
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有些函数在没有拐点的情况下改变凹度。相反，它们可以改变垂直渐近线或不连续性周围的凹度。例如，函数 < math > x mapsto frac1x </math > 是凹的表示负的，凸的表示正的，但是它没有拐点，因为0不在函数的域中。<br />
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== See also ==<br />
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* [[Critical point (mathematics)]]<br />
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* [[Ecological threshold]]<br />
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* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
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* [[Ogee]], an architectural form with an inflection point<br />
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* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
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==References==<br />
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{{reflist}}<br />
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==Sources==<br />
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* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E6%8B%90%E7%82%B9&diff=19679拐点2020-12-05T10:45:27Z<p>Typhoid：</p>
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<div>此词条由Typhoid翻译<br />
{{short description|Point where the curvature of a curve changes sign}}<br />
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{{More footnotes|date=July 2013}}<br />
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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''<sup>3</sup>}} with an inflection point at (0,0), which is also a [[stationary point]].]]<br />
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Plot of with an inflection point at (0,0), which is also a [[stationary point.]]<br />
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拐点在(0,0) ，这也是一个[[驻点]]<br />
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{{Cubic graph special points.svg}}<br />
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In [[differential calculus]] and differential geometry, an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[plane curve#Smooth plane curve|smooth plane curve]] at which the [[signed curvature|curvature]] changes sign. In particular, in the case of the [[graph of a function]], it is a point where the function changes from being [[Concave function|concave]] (concave downward) to [[convex function|convex]] (concave upward), or vice versa.<br />
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In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.<br />
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在微分和微分几何，一个拐点，点的拐点，弯曲，或屈折(英式英语: 拐点)是一个光滑的平面曲线上的一个点，曲率改变符号。特别是，在函数图像的情况下，它是一个点的功能变化从凹(向下)到凸(向上) ，或反之亦然。<br />
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For example, if the curve is the graph of a function {{math|1=''y'' = ''f''(''x'')}}, of [[differentiability class]] {{math|''C''<sup>2</sup>}}, an inflection point of the curve is where ''f<nowiki>''</nowiki>'', the [[second derivative]] of {{mvar|f}}, vanishes (''f<nowiki>''</nowiki> = 0'') and changes its sign at the point (from positive to negative or from negative to positive).<ref>{{Cite book|last=Stewart|first=James|title=Calculus|publisher=Cengage Learning|year=2015|isbn=978-1-285-74062-1|edition=8|location=Boston|pages=281}}</ref> A point where the second derivative vanishes but does not change its sign is sometimes called a '''point of undulation''' or '''undulation point'''.<br />
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For example, if the curve is the graph of a function , of differentiability class , an inflection point of the curve is where f<nowiki></nowiki>, the second derivative of , vanishes (f<nowiki></nowiki> = 0) and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.<br />
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例如，如果曲线是可导性类的函数图像，那么曲线的一个拐点就是 f < nowiki > </nowiki > 的二阶导数，消失了(f < nowiki > </nowiki > = 0)并且改变了它的符号(从正到负或从负到正)。二阶导数消失但其符号不变的点有时称为波动点或波动点。<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a [[regular point of an algebraic variety|regular point]] where the tangent meets the curve to [[Glossary of classical algebraic geometry#O|order]] at least 3, and an undulation point or '''hyperflex''' is defined as a point where the tangent meets the curve to order at least 4.<br />
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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.<br />
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在代数几何中，拐点的定义更为广泛一些，如切线与曲线相交处的正则点至少为3，波动点或超高频点则定义为切线与曲线相交处至少为4。<br />
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==Definition==<br />
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Inflection points in differential geometry are the points of the curve where the [[curvature]] changes its sign.<ref>{{Cite book|title=Problems in mathematical analysis|origyear=1964 |year=1976|publisher=Mir Publishers|others=Baranenkov, G. S.|isbn=5030009434|location=Moscow|oclc=21598952}}</ref><ref>{{cite book |last=Bronshtein |last2=Semendyayev |title=Handbook of Mathematics |edition=4th |location=Berlin |publisher=Springer |year=2004 |isbn=3-540-43491-7 |page=231 }}</ref><br />
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Inflection points in differential geometry are the points of the curve where the curvature changes its sign.<br />
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微分几何中的拐点是曲率改变其符号的曲线的拐点。<br />
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For example, the graph of the [[differentiable function]] has an inflection point at {{math|(''x'', ''f''(''x''))}} if and only if its [[derivative|first derivative]], {{mvar|f'}}, has an [[isolated point|isolated]] [[extremum]] at {{mvar|x}}. (This is not the same as saying that {{mvar|f}} has an extremum). That is, in some neighborhood, {{mvar|x}} is the one and only point at which {{mvar|f'}} has a (local) minimum or maximum. If all [[extremum|extrema]] of {{mvar|f'}} are [[isolated point|isolated]], then an inflection point is a point on the graph of {{mvar|f}} at which the [[tangent]] crosses the curve.<br />
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For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum. If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve.<br />
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例如，可微函数图的一阶导数------------------------------------------------------------------------ -。(这不等于说有极值)。也就是说，在某些邻域中，是唯一一个具有(局部)最小值或最大值的点。如果所有的极值都是孤立的，那么拐点就是曲线图上的一个点，在这个点上切线与曲线相交。<br />
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A ''falling point of inflection'' is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A ''rising point of inflection'' is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.<br />
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拐点的下降点是一个拐点，在这个点的两边导数都是负数; 换句话说，它是一个拐点，在这个点附近函数正在减少。拐点的上升点是一个导数在点的两边都是正的点; 换句话说，它是一个拐点，在这个点附近函数正在增加。<br />
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For an [[algebraic curve]], a non singular point is an inflection point if and only if the [[intersection number]] of the tangent line and the curve (at the point of tangency) is greater than 2.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Point_of_inflection|title=Point of inflection|encyclopedia=encyclopediaofmath.org}}</ref><br />
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For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.<br />
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对于一条代数曲线，一个非奇点是拐点，当且仅当切线与曲线(在切点处)的交点数大于2。<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the [[Polar curve|Hessian curve]].<br />
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The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve.<br />
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主要结果是代数曲线拐点的集合与曲线与黑森曲线的交集相一致。<br />
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For a smooth curve given by [[parametric equation]]s, a point is an inflection point if its [[Curvature#Signed curvature|signed curvature]] changes from plus to minus or from minus to plus, i.e., changes [[sign (mathematics)|sign]].<br />
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For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.<br />
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对于由参数方程组给出的光滑曲线，如果有符号的曲率从正变为负或从负变为正，即改变符号，那么该点就是拐点。<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the [[second derivative]] has an isolated zero and changes sign.<br />
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For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.<br />
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对于一条光滑曲线，它是一个二次可微函数的图形，一个二次拐点是图形上的一个点，在这个点上二次导数有一个孤立的零并且改变符号。<br />
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[[Image:Animated illustration of inflection point.gif|upright=2.5|thumb|Plot of {{math|''f''(''x'') {{=}} sin(2''x'')}} from −{{pi}}/4 to 5{{pi}}/4; the second [[derivative]] is {{math|''f{{''}}''(''x'') {{=}} –4sin(2''x'')}}, and its sign is thus the opposite of the sign of {{mvar|f}}. Tangent is blue where the curve is [[convex function|convex]] (above its own [[tangent line|tangent]]), green where concave (below its tangent), and red at inflection points: 0, {{pi}}/2 and {{pi}}]]<br />
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Plot of sin(2x)}} from −/4 to 5/4; the second [[derivative is (x) –4sin(2x)}}, and its sign is thus the opposite of the sign of . Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, /2 and ]]<br />
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从-/4到5/4的 sin (2x)}的图; 第二个[[导数是(x)-4sin (2x)}] ，它的符号因此与。切线是蓝色的，这里曲线是凸的(在它自己的切线之上) ，绿色的是凹的(在它的切线之下) ，红色的是在拐点: 0,/2和]<br />
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==A necessary but not sufficient condition==<br />
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If the second derivative, {{math|''f{{''}}''(''x'')}} exists at {{math|''x''<sub>0</sub>}}, and {{math|''x''<sub>0</sub>}} is an inflection point for {{mvar|f}}, then {{math|''f{{''}}''(''x''<sub>0</sub>) {{=}} 0}}, but this condition is not [[Sufficient condition|sufficient]] for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an ''undulation point''. However, in algebraic geometry, both inflection points and undulation points are usually called ''inflection points''. An example of an undulation point is {{math|''x'' {{=}} 0}} for the function {{mvar|f}} given by {{math|''f''(''x'') {{=}} ''x''<sup>4</sup>}}.<br />
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If the second derivative, (x)}} exists at , and is an inflection point for , then (x<sub>0</sub>) 0}}, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is 0}} for the function given by x<sup>4</sup>}}.<br />
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如果二阶导数，(x)}}存在于，并且是拐点，那么(x < sub > 0 </sub >)0} ，但是这个条件对于有拐点是不充分的，即使存在任意阶的导数。在这种情况下，还需要最低阶(第二阶以上)非零导数为奇数阶(第三阶、第五阶等)。如果最低阶非零导数为偶数阶，则该点不是拐点，而是波动点。然而，在代数几何中，拐点和起伏点通常都被称为拐点。对于 x < sup > 4 </sup > }给出的函数，波动点的一个例子是0}}。<br />
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In the preceding assertions, it is assumed that {{mvar|f}} has some higher-order non-zero derivative at {{mvar|x}}, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of {{math|''f{{'}}''(''x'')}} is the same on either side of {{mvar|x}} in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}. If this sign is [[positive number|positive]], the point is a ''rising point of inflection''; if it is [[negative number|negative]], the point is a ''falling point of inflection''.<br />
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In the preceding assertions, it is assumed that has some higher-order non-zero derivative at , which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of (x)}} is the same on either side of in a neighborhood of . If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.<br />
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在前面的断言中，假定有一些高阶非零导数在，这不一定是这种情况。如果是这样的话，第一个非零导数有奇数阶的条件意味着(x)}的符号在一个邻域的任一边都是相同的。如果这个符号是正的，那么这个点就是拐点的上升点; 如果是负的，那么这个点就是拐点的下降点。<br />
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'''Inflection points sufficient conditions:'''<br />
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Inflection points sufficient conditions:<br />
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拐点充分条件:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1) A sufficient existence condition for a point of inflection is:<br />
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1)拐点存在的一个充分条件是:<br />
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:If {{math|''f''(''x'')}} is {{mvar|k}} times continuously differentiable in a certain neighborhood of a point {{mvar|x}} with {{mvar|k}} odd and {{math|''k'' ≥ 3}}, while {{math|''f''<sup>(''n'')</sup>(''x''<sub>0</sub>)&nbsp;{{=}}&nbsp;0}} for {{math|''n'' {{=}} 2,&nbsp;&hellip;,&nbsp;''k''&nbsp;−&nbsp;1}} and {{math|''f''<sup>(''k'')</sup>(''x''<sub>0</sub>)&nbsp;≠&nbsp;0}} then {{math|''f''(''x'')}} has a point of inflection at {{math|''x''<sub>0</sub>}}.<br />
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If is times continuously differentiable in a certain neighborhood of a point with odd and , while &nbsp;0}} for 2,&nbsp;&hellip;,&nbsp;k&nbsp;−&nbsp;1}} and then has a point of inflection at .<br />
<br />
如果在一个奇数点和，而0}为2，& hellip; ，k-1}的点的某个邻域内是时间连续可微的，那么在。<br />
<br />
<br />
<br />
2) Another sufficient existence condition requires {{math|''f{{''}}''(''x'' + ε)}} and {{math|''f{{''}}''(''x''&nbsp;−&nbsp;''ε'')}} to have opposite signs in the neighborhood of&nbsp;''x'' ([[Bronshtein and Semendyayev]] 2004, p.&nbsp;231).<br />
<br />
2) Another sufficient existence condition requires (x + ε)}} and (x&nbsp;−&nbsp;ε)}} to have opposite signs in the neighborhood of&nbsp;x (Bronshtein and Semendyayev 2004, p.&nbsp;231).<br />
<br />
2)另一个充分存在条件要求(x + ε)}和(x-ε)}}在 x (Bronshtein 和 Semendyayev，2004，p. 231)附近有相反的符号。<br />
<br />
<br />
<br />
==Categorization of points of inflection==<br />
<br />
[[Image:X to the 4th minus x.svg|thumb|upright=1.2|{{math|''y'' {{=}} ''x''<sup>4</sup> – ''x''}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).]]<br />
<br />
x<sup>4</sup> – x}} has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).<br />
<br />
X < sup > 4 </sup >-x }在点(0,0)处有一个零的二阶导数，但它不是拐点，因为四阶导数是一阶非零导数(三阶导数也是零)。<br />
<br />
<br />
<br />
Points of inflection can also be categorized according to whether {{math|''f{{'}}''(''x'')}} is zero or nonzero.<br />
<br />
Points of inflection can also be categorized according to whether (x)}} is zero or nonzero.<br />
<br />
拐点也可以根据(x)}是零还是非零来分类。<br />
<br />
* if {{math|''f{{'}}''(''x'')}} is zero, the point is a ''[[stationary point]] of inflection''<br />
<br />
* if {{math|''f{{'}}''(''x'')}} is not zero, the point is a ''non-stationary point of inflection''<br />
<br />
<br />
<br />
A stationary point of inflection is not a [[local extremum]]. More generally, in the context of [[functions of several real variables]], a stationary point that is not a local extremum is called a [[saddle point#Mathematical discussion|saddle point]].<br />
<br />
A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.<br />
<br />
一个驻点的曲折变化并不是局部极值。更广泛地说，在多个实变量函数的背景下，一个不是局部极值的驻点被称为鞍点。<br />
<br />
<br />
<br />
An example of a stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup>}}. The tangent is the {{mvar|x}}-axis, which cuts the graph at this point.<br />
<br />
An example of a stationary point of inflection is the point on the graph of x<sup>3</sup>}}. The tangent is the -axis, which cuts the graph at this point.<br />
<br />
一个驻点拐点的例子是 x < sup > 3 </sup > }图上的点。切线是-轴，它在此处切割图形。<br />
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<br />
<br />
An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''<sup>3</sup> + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.<br />
<br />
An example of a non-stationary point of inflection is the point on the graph of x<sup>3</sup> + ax}}, for any nonzero . The tangent at the origin is the line ax}}, which cuts the graph at this point.<br />
<br />
非平稳拐点的一个例子是 x < sup > 3 </sup > + ax }图上对任意非零点的拐点。原点处的切线是直线 ax }}}} ，它在此处切割图形。<br />
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<br />
<br />
==Functions with discontinuities==<br />
<br />
Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.<br />
<br />
Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.<br />
<br />
有些函数在没有拐点的情况下改变凹度。相反，它们可以改变垂直渐近线或不连续性周围的凹度。例如，函数 < math > x mapsto frac1x </math > 是凹的表示负的，凸的表示正的，但是它没有拐点，因为0不在函数的域中。<br />
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<br />
== See also ==<br />
<br />
* [[Critical point (mathematics)]]<br />
<br />
* [[Ecological threshold]]<br />
<br />
* [[Hesse configuration]] formed by the nine inflection points of an [[elliptic curve]]<br />
<br />
* [[Ogee]], an architectural form with an inflection point<br />
<br />
* [[Vertex (curve)]], a local minimum or maximum of curvature<br />
<br />
<br />
<br />
==References==<br />
<br />
{{reflist}}<br />
<br />
<br />
<br />
==Sources==<br />
<br />
* {{MathWorld|title=Inflection Point|urlname=InflectionPoint}}<br />
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* {{springer|title=Point of inflection|id=p/p073190}}<br />
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[[Category:Differential calculus]]<br />
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Category:Differential calculus<br />
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类别: 微分<br />
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[[Category:Curves]]<br />
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Category:Curves<br />
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类别: 曲线<br />
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[[Category:Analytic geometry]]<br />
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Category:Analytic geometry<br />
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类别: 解析几何<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Inflection point]]. Its edit history can be viewed at [[拐点/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E5%B1%82%E7%BA%A7%E7%90%86%E8%AE%BA&diff=18799层级理论2020-11-19T23:53:03Z<p>Typhoid：/* See also */</p>
<hr />
<div>此词条由Typhoid翻译<br />
<br />
'''Hierarchy theory''' is a means of studying [[ecological systems]] in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on [[Level of organization|levels of organization]] and issues of [[Scale (analytical tool)|scale]], with a specific focus on the role of the observer in the definition of the system.<ref>{{cite web |last=Allen |first=Timothy F. H. |date=2001 |title=A summary of the principles of hierarchy theory |url=http://isss.org/hierarchy.htm |archiveurl=https://web.archive.org/web/20011218001638/http://isss.org/hierarchy.htm |archivedate=2001-12-18 |accessdate=2016-03-19}}</ref> Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
Hierarchy theory is a means of studying ecological systems in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on levels of organization and issues of scale, with a specific focus on the role of the observer in the definition of the system. Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
层级理论是当各组成部分存在很强的复杂度关系时研究生态系统的一种方法。层级理论聚焦于层次结构与规模问题，具体针对于观察者在系统定义中的作用。在这种情况下，复杂度不是指代系统的一种固有属性，而是指根据观察者的预分析选择，以多元不等价的方式来描绘系统的可能性。层级理论指的是对层次结构与其间相互作用的分析，而不是去分析整个结构。<br />
<br />
--[[用户:Typhoid|Typhoid]]（[[用户讨论:Typhoid|讨论]]）[翻译] 存疑“在这种情况下，复杂度不是指代系统的一种固有属性，而是指根据观察者的预分析选择，以多元不等价的方式来描绘系统的可能性。”<br />
<br />
==See also==<br />
<br />
* [[Biological organisation]] 生物学组织<br />
<br />
* [[Timothy F. H. Allen]] 蒂莫西．艾伦<br />
<br />
* [[Deep history]] 深历史<br />
<br />
* [[Big history]] 大历史<br />
<br />
* [[Deep time]] 深时<br />
<br />
* [[Deep ecology]] 深层生态学<br />
<br />
* [[Infrastructure-based development]] 轨道发展<br />
<br />
* [[World-systems theory]] 世界系统理论<br />
<br />
* [[Structuralist economics]] 结构性经济<br />
<br />
* [[Dependency theory]] 独立理论<br />
<br />
==References==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==Further reading==<br />
<br />
* {{cite thesis |type=Ph.D. thesis |last=Brooks |first=Daniel Stephen |date=August 2014 |title=The concept of levels of organization in the biological sciences |location=Bielefeld |publisher=[[Bielefeld University]] |oclc=942715109 |url=http://d-nb.info/1082033960/34 |ref=harv}}<br />
<br />
* {{cite journal |last=Eronen |first=Markus I. |date=August 2014 |title=Levels of organization: a deflationary account |journal=[[Biology and Philosophy]] |volume=30 |issue=1 |pages=39–58 |doi=10.1007/s10539-014-9461-z |ref=harv}}<br />
<br />
* {{cite journal |last1=Potochnik |first1=Angela |last2=McGill |first2=Brian J. |date=January 2012 |title=The limitations of hierarchical organization |journal=[[Philosophy of Science (journal)|Philosophy of Science]] |volume=79 |issue=1 |pages=120–140 |doi=10.1086/663237 |jstor=663237 |url=http://homepages.uc.edu/~potochaa/hierarchy.pdf |ref=harv}}<br />
<br />
* {{cite book |last1=Ahl |first1=Valerie |last2=Allen |first2=Timothy F. H. |date=1996 |title=Hierarchy theory: a vision, vocabulary, and epistemology |location=New York |publisher=[[Columbia University Press]] |isbn=0231084803 |oclc=34149766 |ref=harv}}<br />
<br />
* {{cite book |last1=Allen |first1=Timothy F. H. |last2=Hoekstra |first2=Thomas W. |date=2015 |origyear=1992 |title=Toward a unified ecology |edition=2nd |series=Complexity in ecological systems series |location=New York |publisher=[[Columbia University Press]] |isbn=9780231168885 |oclc=920475391 |ref=harv}}<br />
<br />
* {{cite book |last1=O'Neill |first1=Robert V. |last2=Deangelis |first2=Donald Lee |last3=Waide |first3=J. B. |last4=Allen |first4=Timothy F. H. |date=1986 |title=A hierarchical concept of ecosystems |series=Monographs in population biology |volume=23 |location=Princeton, NJ |publisher=[[Princeton University Press]] |isbn=069108436X |oclc=13526197 |url=https://books.google.com/books?id=DOU9DwAAQBAJ |ref=harv}}<br />
<br />
* {{cite book |last1=Allen |first1=Timothy F. H |last2=Starr |first2=Thomas B. |date=2017 |origyear=1982 |title=Hierarchy: perspectives for ecological complexity |edition=2nd |location=Chicago |publisher=[[University of Chicago Press]] |isbn=978-0226489544 |oclc=967919711 |doi=10.7208/chicago/9780226489711.001.0001 |ref=harv}}<br />
<br />
* {{cite journal |last=Guttman |first=Burton S. |date=February 1976 |title=Is 'levels of organization' a useful biological concept? |journal=[[BioScience]] |volume=26 |issue=2 |pages=112–113 |doi=10.2307/1297326 |jstor=1297326 |ref=harv}}<br />
<br />
* {{cite book |last=Pattee |first=Howard Hunt |authorlink=Howard H. Pattee |date=1973 |title=Hierarchy theory: the challenge of complex systems |series=International library of systems theory and philosophy |location=New York |publisher=[[George Braziller]] |isbn=080760674X |oclc=638741 |ref=harv}}<br />
<br />
<br />
<br />
[[Category:Hierarchy]]<br />
<br />
Category:Hierarchy<br />
<br />
类别: 层级<br />
<br />
[[Category:Systems ecology]]<br />
<br />
Category:Systems ecology<br />
<br />
类别: 系统生态学<br />
<br />
<br />
<br />
<br />
<br />
{{Sci-stub}}<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Hierarchy theory]]. Its edit history can be viewed at [[层级理论/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E5%B1%82%E7%BA%A7%E7%90%86%E8%AE%BA&diff=18126层级理论2020-11-11T16:24:42Z<p>Typhoid：</p>
<hr />
<div>此词条由Typhoid翻译<br />
<br />
'''Hierarchy theory''' is a means of studying [[ecological systems]] in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on [[Level of organization|levels of organization]] and issues of [[Scale (analytical tool)|scale]], with a specific focus on the role of the observer in the definition of the system.<ref>{{cite web |last=Allen |first=Timothy F. H. |date=2001 |title=A summary of the principles of hierarchy theory |url=http://isss.org/hierarchy.htm |archiveurl=https://web.archive.org/web/20011218001638/http://isss.org/hierarchy.htm |archivedate=2001-12-18 |accessdate=2016-03-19}}</ref> Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
Hierarchy theory is a means of studying ecological systems in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on levels of organization and issues of scale, with a specific focus on the role of the observer in the definition of the system. Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
层级理论是当各组成部分存在很强的复杂度关系时研究生态系统的一种方法。层级理论聚焦于层次结构与规模问题，具体针对于观察者在系统定义中的作用。在这种情况下，复杂度不是指代系统的一种固有属性，而是指根据观察者的预分析选择，以多元不等价的方式来描绘系统的可能性。层级理论指的是对层次结构与其间相互作用的分析，而不是去分析整个结构。<br />
<br />
--[[用户:Typhoid|Typhoid]]（[[用户讨论:Typhoid|讨论]]）[翻译] 存疑“在这种情况下，复杂度不是指代系统的一种固有属性，而是指根据观察者的预分析选择，以多元不等价的方式来描绘系统的可能性。”<br />
<br />
==See also==<br />
<br />
* [[Biological organisation]]<br />
<br />
* [[Timothy F. H. Allen]]<br />
<br />
* [[Deep history]]<br />
<br />
* [[Big history]]<br />
<br />
* [[Deep time]]<br />
<br />
* [[Deep ecology]]<br />
<br />
* [[Infrastructure-based development]]<br />
<br />
* [[World-systems theory]]<br />
<br />
* [[Structuralist economics]]<br />
<br />
* [[Dependency theory]]<br />
<br />
<br />
<br />
==References==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==Further reading==<br />
<br />
* {{cite thesis |type=Ph.D. thesis |last=Brooks |first=Daniel Stephen |date=August 2014 |title=The concept of levels of organization in the biological sciences |location=Bielefeld |publisher=[[Bielefeld University]] |oclc=942715109 |url=http://d-nb.info/1082033960/34 |ref=harv}}<br />
<br />
* {{cite journal |last=Eronen |first=Markus I. |date=August 2014 |title=Levels of organization: a deflationary account |journal=[[Biology and Philosophy]] |volume=30 |issue=1 |pages=39–58 |doi=10.1007/s10539-014-9461-z |ref=harv}}<br />
<br />
* {{cite journal |last1=Potochnik |first1=Angela |last2=McGill |first2=Brian J. |date=January 2012 |title=The limitations of hierarchical organization |journal=[[Philosophy of Science (journal)|Philosophy of Science]] |volume=79 |issue=1 |pages=120–140 |doi=10.1086/663237 |jstor=663237 |url=http://homepages.uc.edu/~potochaa/hierarchy.pdf |ref=harv}}<br />
<br />
* {{cite book |last1=Ahl |first1=Valerie |last2=Allen |first2=Timothy F. H. |date=1996 |title=Hierarchy theory: a vision, vocabulary, and epistemology |location=New York |publisher=[[Columbia University Press]] |isbn=0231084803 |oclc=34149766 |ref=harv}}<br />
<br />
* {{cite book |last1=Allen |first1=Timothy F. H. |last2=Hoekstra |first2=Thomas W. |date=2015 |origyear=1992 |title=Toward a unified ecology |edition=2nd |series=Complexity in ecological systems series |location=New York |publisher=[[Columbia University Press]] |isbn=9780231168885 |oclc=920475391 |ref=harv}}<br />
<br />
* {{cite book |last1=O'Neill |first1=Robert V. |last2=Deangelis |first2=Donald Lee |last3=Waide |first3=J. B. |last4=Allen |first4=Timothy F. H. |date=1986 |title=A hierarchical concept of ecosystems |series=Monographs in population biology |volume=23 |location=Princeton, NJ |publisher=[[Princeton University Press]] |isbn=069108436X |oclc=13526197 |url=https://books.google.com/books?id=DOU9DwAAQBAJ |ref=harv}}<br />
<br />
* {{cite book |last1=Allen |first1=Timothy F. H |last2=Starr |first2=Thomas B. |date=2017 |origyear=1982 |title=Hierarchy: perspectives for ecological complexity |edition=2nd |location=Chicago |publisher=[[University of Chicago Press]] |isbn=978-0226489544 |oclc=967919711 |doi=10.7208/chicago/9780226489711.001.0001 |ref=harv}}<br />
<br />
* {{cite journal |last=Guttman |first=Burton S. |date=February 1976 |title=Is 'levels of organization' a useful biological concept? |journal=[[BioScience]] |volume=26 |issue=2 |pages=112–113 |doi=10.2307/1297326 |jstor=1297326 |ref=harv}}<br />
<br />
* {{cite book |last=Pattee |first=Howard Hunt |authorlink=Howard H. Pattee |date=1973 |title=Hierarchy theory: the challenge of complex systems |series=International library of systems theory and philosophy |location=New York |publisher=[[George Braziller]] |isbn=080760674X |oclc=638741 |ref=harv}}<br />
<br />
<br />
<br />
[[Category:Hierarchy]]<br />
<br />
Category:Hierarchy<br />
<br />
类别: 层级<br />
<br />
[[Category:Systems ecology]]<br />
<br />
Category:Systems ecology<br />
<br />
类别: 系统生态学<br />
<br />
<br />
<br />
<br />
<br />
{{Sci-stub}}<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Hierarchy theory]]. Its edit history can be viewed at [[层级理论/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Typhoidhttps://wiki.swarma.org/index.php?title=%E5%B1%82%E7%BA%A7%E7%90%86%E8%AE%BA&diff=18125层级理论2020-11-11T16:11:04Z<p>Typhoid：</p>
<hr />
<div>此词条暂由彩云小译翻译，翻译字数共108，未经人工整理和审校，带来阅读不便，请见谅。<br />
<br />
'''Hierarchy theory''' is a means of studying [[ecological systems]] in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on [[Level of organization|levels of organization]] and issues of [[Scale (analytical tool)|scale]], with a specific focus on the role of the observer in the definition of the system.<ref>{{cite web |last=Allen |first=Timothy F. H. |date=2001 |title=A summary of the principles of hierarchy theory |url=http://isss.org/hierarchy.htm |archiveurl=https://web.archive.org/web/20011218001638/http://isss.org/hierarchy.htm |archivedate=2001-12-18 |accessdate=2016-03-19}}</ref> Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
Hierarchy theory is a means of studying ecological systems in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on levels of organization and issues of scale, with a specific focus on the role of the observer in the definition of the system. Complexity in this context does not refer to an intrinsic property of the system but to the possibility of representing the systems in a plurality of non-equivalent ways depending on the pre-analytical choices of the observer. Instead of analyzing the whole structure, hierarchy theory refers to the analysis of hierarchical levels, and the interactions between them.<br />
<br />
层级理论是当各组成部分存在很强的复杂度关系时研究生态系统的一种方法。层级理论聚焦于层次结构与规模问题，具体针对于观察者在系统定义中的作用。在这种情况下，复杂度不是指代系统的一种固有属性，而是指根据观察者的预分析选择，以多元不等价的方式来描绘系统的可能性。层级理论指的是对层次结构与其间相互作用的分析，而不是去分析整个结构。<br />
<br />
<br />
<br />
==See also==<br />
<br />
* [[Biological organisation]]<br />
<br />
* [[Timothy F. H. Allen]]<br />
<br />
* [[Deep history]]<br />
<br />
* [[Big history]]<br />
<br />
* [[Deep time]]<br />
<br />
* [[Deep ecology]]<br />
<br />
* [[Infrastructure-based development]]<br />
<br />
* [[World-systems theory]]<br />
<br />
* [[Structuralist economics]]<br />
<br />
* [[Dependency theory]]<br />
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==References==<br />
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{{Reflist}}<br />
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==Further reading==<br />
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* {{cite thesis |type=Ph.D. thesis |last=Brooks |first=Daniel Stephen |date=August 2014 |title=The concept of levels of organization in the biological sciences |location=Bielefeld |publisher=[[Bielefeld University]] |oclc=942715109 |url=http://d-nb.info/1082033960/34 |ref=harv}}<br />
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* {{cite journal |last=Eronen |first=Markus I. |date=August 2014 |title=Levels of organization: a deflationary account |journal=[[Biology and Philosophy]] |volume=30 |issue=1 |pages=39–58 |doi=10.1007/s10539-014-9461-z |ref=harv}}<br />
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* {{cite journal |last1=Potochnik |first1=Angela |last2=McGill |first2=Brian J. |date=January 2012 |title=The limitations of hierarchical organization |journal=[[Philosophy of Science (journal)|Philosophy of Science]] |volume=79 |issue=1 |pages=120–140 |doi=10.1086/663237 |jstor=663237 |url=http://homepages.uc.edu/~potochaa/hierarchy.pdf |ref=harv}}<br />
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* {{cite book |last1=Ahl |first1=Valerie |last2=Allen |first2=Timothy F. H. |date=1996 |title=Hierarchy theory: a vision, vocabulary, and epistemology |location=New York |publisher=[[Columbia University Press]] |isbn=0231084803 |oclc=34149766 |ref=harv}}<br />
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* {{cite book |last1=Allen |first1=Timothy F. H. |last2=Hoekstra |first2=Thomas W. |date=2015 |origyear=1992 |title=Toward a unified ecology |edition=2nd |series=Complexity in ecological systems series |location=New York |publisher=[[Columbia University Press]] |isbn=9780231168885 |oclc=920475391 |ref=harv}}<br />
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* {{cite book |last1=O'Neill |first1=Robert V. |last2=Deangelis |first2=Donald Lee |last3=Waide |first3=J. B. |last4=Allen |first4=Timothy F. H. |date=1986 |title=A hierarchical concept of ecosystems |series=Monographs in population biology |volume=23 |location=Princeton, NJ |publisher=[[Princeton University Press]] |isbn=069108436X |oclc=13526197 |url=https://books.google.com/books?id=DOU9DwAAQBAJ |ref=harv}}<br />
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* {{cite book |last1=Allen |first1=Timothy F. H |last2=Starr |first2=Thomas B. |date=2017 |origyear=1982 |title=Hierarchy: perspectives for ecological complexity |edition=2nd |location=Chicago |publisher=[[University of Chicago Press]] |isbn=978-0226489544 |oclc=967919711 |doi=10.7208/chicago/9780226489711.001.0001 |ref=harv}}<br />
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* {{cite journal |last=Guttman |first=Burton S. |date=February 1976 |title=Is 'levels of organization' a useful biological concept? |journal=[[BioScience]] |volume=26 |issue=2 |pages=112–113 |doi=10.2307/1297326 |jstor=1297326 |ref=harv}}<br />
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* {{cite book |last=Pattee |first=Howard Hunt |authorlink=Howard H. Pattee |date=1973 |title=Hierarchy theory: the challenge of complex systems |series=International library of systems theory and philosophy |location=New York |publisher=[[George Braziller]] |isbn=080760674X |oclc=638741 |ref=harv}}<br />
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[[Category:Hierarchy]]<br />
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Category:Hierarchy<br />
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类别: 层级<br />
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[[Category:Systems ecology]]<br />
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Category:Systems ecology<br />
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类别: 系统生态学<br />
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{{Sci-stub}}<br />
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<small>This page was moved from [[wikipedia:en:Hierarchy theory]]. Its edit history can be viewed at [[层级理论/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Typhoid