About: Principal curvature

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In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.

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rdf:type	yago:Artifact100021939 yago:Object100002684 yago:PhysicalEntity100001930 yago:Surface104362025 yago:Whole100003553 yago:WikicatSurfaces
rdfs:label	انحناءات رئيسية (ar) Hauptkrümmung (de) Courbure principale (fr) Curvatura principale (it) 主曲率 (ja) 주곡률 (ko) Hoofdkromming (nl) Principal curvature (en) Krzywizna główna (pl) 主曲率 (zh)
rdfs:comment	االانحناءين الرئيسيين (Principal curvature) لسطح س في الفضاء الإقليدي يشيران إلى الحد الأقصى والحد الأدنى لانحناء س في نقطة منه.يتم الحصول على كمتوسط حسابي للانحناءين الرئيسيين. (ar) Hauptkrümmung ist ein Begriff aus der Differentialgeometrie. Jedem Punkt einer Fläche im dreidimensionalen euklidischen Raum werden zwei Hauptkrümmungen zugeordnet. (de) In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point. (en) 微分幾何学において、曲面上の与えられた点での 2つの主曲率(英: principal curvature)は、その点でのガウス写像(Gauss map)の微分の 2つの固有値である。それらは、曲面がその点で別々な方向へどれくらい曲がっているかを測る。 (ja) In geometria differenziale, ad ogni punto di una superficie differenziabile nello spazio euclideo sono associate due curvature principali: queste sono il massimo ed il minimo della curvatura di una curva contenuta nella superficie e passante per il punto. La curvatura gaussiana e la curvatura media sono ottenute rispettivamente come prodotto e come media aritmetica delle due curvature principali. (it) 미분기하학에서 주곡률(主曲率, 영어: principal curvature)은 곡면이 각 방향에 따라 굽은 정도를 나타내는 값들이며, 제2 기본 형식의 고윳값들이다. (ko) Krzywizny główne – rodzaj miary zakrzywienia powierzchni dwuwymiarowej. (pl) 在微分几何中，在曲面给定点的两个主曲率（principal curvatures）衡量了在给定点一个曲面在这一点的不同方向怎样不同弯曲的程度。在曲面上取一点E，曲面在E点的法线为z轴，过z轴可以有无限多个剖切平面，每个剖切平面与曲面相交，其交线为一条平面曲线，每条平面曲线在E点有一个曲率半径。不同的剖切平面上的平面曲线在E点的曲率半径一般是不相等的。这些曲率半径中，有一个最大和最小的曲率半径，称之为主曲率半径，记作 k1 与 k2，这两个曲率半径所在的方向，数学上可以证明是相互垂直的。这里一条曲线的曲率由定义是半径的倒数。当曲线转向与平面给定法向量相同方向时，曲率取正值，否则取负值。当曲率取最大与最小值的两个法平面方向总是垂直的，这是欧拉在1760年的一个结论，称之为主方向。从现代的观点来看，这个定理来自谱定理因为它们可以作为对应于高斯映射微分的一个对称矩阵的本征向量。对主曲率和主方向的系统研究由达布使用完成。两个主曲率的乘积 k1k2 是高斯曲率 K，而平均值 (k1+k2)/2 是平均曲率 H。如果在每一点至少有一个主曲率是零，则高斯曲率是零，这种曲面是可展曲面。对极小曲面，平均曲率在每一点是零。 (zh) En géométrie différentielle des surfaces, les deux courbures principales d'une surface sont les courbures de cette surface selon deux directions perpendiculaires appelées directions principales. On montre que ce sont les courbures minimale et maximale rencontrées en faisant tourner le plan de coupe. Les courbures principales sont les valeurs propres de l'endomorphisme de Weingarten. Elles caractérisent la géométrie locale des surfaces à l'ordre 2. (fr) In de differentiaalmeetkunde, een deelgebied van de meetkunde, meten de twee hoofdkrommingen op een bepaald punt van een oppervlak, hoe dit oppervlak op dat punt in verschillende richtingen in verschillende mate buigt. De twee zo bepaalde normale vlakken met een hoofdkromming staan altijd loodrecht op elkaar. (nl)
foaf:depiction
dcterms:subject	Surfaces Curvature (mathematics) Differential geometry of surfaces
Wikipage page ID	2072472 (xsd:integer)
Wikipage revision ID	1106254549 (xsd:integer)
Link from a Wikipage to another Wikipage	Normal vector Surfaces Developable surface Integral curve Multiplicative inverse Normal plane (geometry) Eigenvalue Eigenvector Gaussian curvature Monkey saddle Leonhard Euler Curvature (mathematics) Orthonormal basis Surface (differential geometry) Mean curvature Local property Minimal surface Curvature Earth radius Euclidean space Euler's theorem (differential geometry) Differential geometry of surfaces Osculating circle Darboux frame Principal axis theorem Radius Ridge (differential geometry) Riemannian manifold Surface (mathematics) Symmetric tensor Differentiable manifold Differential geometry Differential geometry of surfaces Spectral theorem Image segmentation Second fundamental form Umbilical point Motion estimation Normal section Sectional curvature Orientation (mathematics) Eigenvalues Gaston Darboux Shape operator dbr:Wikt:convex
Link from a Wikipage to an external page	http://mat.uab.cat/web/matmat/en/v2007n01/ https://rmu.sbm.org.br/wp-content/uploads/sites/27/2018/03/n15_Artigo04.pdf https://archive.org/details/leconsurlagendes01darbrich/page/%7Cpublisher=Gauthier-Villars https://arxiv.org/abs/math/0411403
sameAs	Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature Principal curvature

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