The Four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. The Four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya.

PropertyValue
dbpedia-owl:thumbnail
dbpprop:abstract
  • The Four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. The Four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4th-order contact with the curve (in general the osculating circle has only 3rd-order contact with the curve). The Four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument. The converse to the Four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for strictly positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres. The full converse to the Four-vertex theorem was proved by Björn Dahlberg shortly before his death in January, 1998 and published posthumously. Dahlberg's proof uses a winding number argument which is in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra. One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points. There is no 3-dimensional generalization of this result as there exist a convex, homogeneous object with exactly 2 balance points (not necessarily stable), see Gomboc.
  • Le théorème des 4 sommets consititue un résultat remarquable de géométrie différentielle quant aux propriétés globales des courbes fermées.
  • 四頂點定理是微分幾何關於平面曲線的整體性質的定理。這定理指出，一條簡單閉曲線的曲率函數，如果不是常值，便有至少四個局部極值。更確切地說，這函數有至少兩個局部極大值和兩個局部極小值。 1909年斯亞馬達斯·穆科帕迪亞亞最先證明這定理對凸曲線（即有嚴格正曲率）成立。他的證明用到了以下結果：曲線上一點的曲率是極值，當且僅當在該點的密切圓與曲線有4點切觸。（密切圓與曲線一般只有3點切觸。）1912年阿道夫·克內澤爾證明了定理在一般情況成立。 四頂點定理的逆定理指，在圓上定義任意連續實值函數，使得有兩個局部極大值和兩個局部極小值，那麼這函數是一條簡單平面閉曲線的曲率函數。1971年赫爾曼·格盧克證出嚴格正函數的情形。他證明在n維球面預先定義曲率的更一般定理，以上結果是其特例。比約恩·達爾貝里在他1998年1月去世前不久，證明逆定理的完整版本。他的證法用到卷繞數，類似代數基本定理的拓撲證明。 這定理的一個推論是，任何在平面上滾動受重力作用的均勻板，都有至少四個平衡點。它的三維推廣並不容易，實際上，存在少於四個平衡點的三維凸均勻體，見Gömböc。
dbpprop:hasPhotoCollection
dbpprop:reference
rdf:type
rdfs:comment
  • The Four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. The Four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya.
  • Le théorème des 4 sommets consititue un résultat remarquable de géométrie différentielle quant aux propriétés globales des courbes fermées.
rdfs:label
  • Four-vertex theorem
  • Théorème des quatre sommets
  • 四頂點定理
owl:sameAs
skos:subject
foaf:depiction
foaf:page
is dbpprop:redirect of
is owl:sameAs of