Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry.

Differential geometry

Main article: differential geometry of curves

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While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.

Related Topics:
Helix - Classical mechanics - General relativity - World line - Spacetime

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If X is a differentiable manifold, then we can define the notion of differentiable curve in X. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take X to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X by means of this notion of curve.

Related Topics:
Differentiable manifold - Euclidean space - Tangent vector

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If X is a smooth manifold, a smooth curve in X is a smooth map

Related Topics:
Smooth manifold - Smooth map

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:!,gamma : I ightarrow X.

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This is a basic notion. There are less and more restricted ideas, too. If X is a C^k manifold (i.e., a manifold whose charts are k times continuously differentiable), then a C^k curve in X is such a curve which is only assumed to be C^k (i.e. k times continuously differentiable). If X is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and !,gamma is an analytic map, then !,gamma is said to be an analytic curve.

Related Topics:
Charts - Continuously differentiable - Analytic manifold - Power series

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A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two C^k differentiable curves

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:!,gamma_1 :I ightarrow X and

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:!,gamma_2 : J ightarrow X

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are said to be equivalent if there is a bijective C^k map

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:!,p : J ightarrow I

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such that the inverse map

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:!,p^{-1} : I ightarrow J

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is also C^k, and

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:!,gamma_{2}(t) = gamma_{1}(p(t))

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for all t. The map !,gamma_2 is called a reparametrisation of !,gamma_1; and this makes an equivalence relation on the set of all C^k differentiable curves in X. A C^k arc is an equivalence class of C^k curves under the relation of reparametrisation.

Related Topics:
Equivalence relation - Equivalence class

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