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:''This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds. For a discussion of curves in an arbitrary topological space, see the main article on curves.'' Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus. Starting in antiquity, many concrete 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. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations, because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the ''natural parametrization'') and from the point of view of a bug on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by the way in which they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the ''curvature'' and the ''torsion'' of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. == Definitions == (詳細はnon-empty interval of real numbers and ''t'' in ''I''. A vector-valued function : of class ''C''''r'' (i.e. γ is ''r'' times continuously differentiable) is called a parametric curve of class Cr or a ''C''''r'' parametrization of the curve γ. ''t'' is called the parameter of the curve γ. γ(''I'') is called the image of the curve. It is important to distinguish between a curve γ and the image of a curve γ(''I'') because a given image can be described by several different ''C''''r'' curves. One may think of the parameter ''t'' as representing time and the curve γ(''t'') as the trajectory of a moving particle in space. If ''I'' is a closed interval (''b'' ), we call γ(''a'') the starting point and γ(''b'') the endpoint of the curve γ. If γ(''a'') = γ(''b''), we say γ is closed or a loop. Furthermore, we call γ a closed Cr-curve if γ(''k'')(a) = γ(''k'')(''b'') for all ''k'' ≤ ''r''. If γ: (''a'',''b'') → R''n'' is injective, we call the curve simple. If γ is a parametric curve which can be locally described as a power series, we call the curve analytic or of class . We write -γ to say the curve is traversed in opposite direction. A ''C''''k''-curve : is called regular of order m if for any ''t'' in interval ''I'' : are linearly independent in R''n''. In particular, a ''C''1-curve ''γ'' is regular if : for any . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Differential geometry of curves」の詳細全文を読む スポンサード リンク
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