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Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.

Attributes	Values
rdf:type	Thing
rdfs:label	هندسة المنحنيات التفاضلية (ar) Geometría diferencial de curvas (es) Differentiable curve (en) Geometria differenziale delle curve (it) Curva diferenciável (pt) Дифференциальная геометрия кривых (ru) 曲线的微分几何 (zh) Диференціальна геометрія кривих (uk)
rdfs:comment	En matemáticas, la geometría diferencial de curvas propone definiciones y métodos para analizar curvas simples en Variedades de Riemann, y en particular, en el Espacio Euclídeo. (es) In matematica, la geometria differenziale delle curve usa l'analisi matematica per studiare le curve nel piano, nello spazio e più generalmente in uno spazio euclideo. (it) Дифференциальная геометрия кривых — раздел дифференциальной геометрии, который занимается исследованием гладких пространственных и плоских кривых в евклидовом пространстве аналитическими методами. (ru) 曲线的微分几何是几何学的一个分支，使用微分与积分专门研究平面与欧几里得空间中的光滑曲线。从古代开始，许多已经用综合方法深入研究。微分几何采取另外一种方式：把曲线表示为参数形式，将它们的几何性质和各种量，比如曲率和弧长，用向量分析表示为导数和积分。分析曲线最重要的工具之一为 Frenet 标架，是一个活动标架，在曲线每一点附近给出“最合适”的坐标系。曲线的理论比及其高维推广的范围要狭窄得多，也简单得多。因为欧几里得空间中的正则曲线没有内蕴几何。任何正则曲线可以用弧长（“自然参数”）参数化，从曲线上来看不能知道周围空间的任何信息，所有曲线都是一样的。不同空间曲线只是由它们的弯曲和扭曲程度区分。数量上，这由微分几何不变量曲线的“曲率”和“挠率”来衡量。断言这些不变量的信息完全确定了曲线。 (zh) هندسة المنحنيات التفاضلية (بالإنجليزية: differential geometry of curves)‏ هي فرع من الهندسة يهتم بالمنحنيات الملساء في المستوي والفضاء الاقليدي باستعمال طرائق حسبان التفاضل والتكامل.ابتداء من العصور القديمة، قد حققت العديد من المنحنيات ملموسة بدقة باستخدام نهج الاصطناعية. الهندسة التفاضلية يأخذ طريقا آخر: يتم تمثيل المنحنيات في الصيغة البارامترية، و خصائصها الهندسية و كمياتها المختلفة والمرتبطة بها، مثل الانحناء وطول القوس، و يعبر بها عن طريق المشتقات و التكامل باستعمال حساب التفاضل والتكامل للمتجهات. واحدة من أهم الأدوات المستخدمة لتحليل منحنى هو الإطار Frenet ، إطار التحرك الذي يوفر نظام الإحداثيات في كل نقطة من المنحنى وهذا هو «أفضل تكييفها» ل منحنى قرب تلك النقطة. (ar) Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point. (en) Geometria diferencial de curvas é o campo da geometria que trabalha com curvas suaves no plano e no espaço euclidiano através de métodos de cálculo diferencial e integral. Numerosas curvas específicas foram estudadas rigorosamente usando a abordagem sintética . A geometria diferencial toma outro rumo: as curvas são retratadas em uma forma parametrizada e suas propriedades geométricas e várias quantidades associadas a elas, como a curvatura e o comprimento do arco, são representadas através de derivadas e integrais usando cálculo vetorial . Uma das ferramentas de maior relevância utilizadas para analisar uma curva é o quadro Frenet, sendo esse um quadro em movimento que fornece um sistema de coordenadas em que cada ponto da curva é "ajustado" à ela próximo a esse ponto. (pt) Диференціа́льна геоме́трія криви́х — це розділ геометрії, який має справу з гладкими кривими на площині та у Евклідовому просторі і використовує для цього методи інтегрального та диференціального числення. (uk)
rdfs:seeAlso	Curve Position vector
foaf:depiction
dcterms:subject	Differential geometry Curves
Wikipage page ID	493403 (xsd:integer)
Wikipage revision ID	1100999164 (xsd:integer)
Link from a Wikipage to another Wikipage	List of curves List of curves topics Normal vector Torsion (differential geometry) Torsion of curves Derivative Arc length Curve Vector (geometric) Velocity Multiplicative inverse Test particle Differential geometry Analytic function Orthonormal Third derivative Circle Frenet–Serret formulas Geodesic Geometry Equivalence class Osculating plane Bijective Zero vector Fundamental theorem of curves Parametric equation Action (physics) Curvature Equivalence relation Euclidean space Euler–Lagrange equation Gram–Schmidt process Moving frame Interval (mathematics) Plane curve Vector-valued function Curves Trajectory Differential calculus Differential geometry Differential geometry of surfaces Continuously differentiable Integral Integral calculus Ordinary differential equations Radius of curvature (mathematics) Vector calculus Euclidean group Euclidean plane Linearly independent Spherical image Injective Synthetic geometry Tangent vector Frenet frame Smoothness (mathematics)
sameAs	Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve Differentiable curve
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