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.

Length of curves

If X is a metric space with metric d, then we can define the length of a curve !,gamma : ightarrow X by

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:mbox{Length} (gamma)=sup left{ sum_{i=1}^n d(gamma(t_i),gamma(t_{i-1})) : n in mathbb{N} mbox{ and } a = t_0 < t_1 < dots < t_n = b ight}

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A rectifiable curve is a curve with finite length.

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A parametrization of !,gamma is called natural (or unit speed or parametrised by arc length) if for any t_1, t_2 in , we have

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: mbox{Length} (gamma|_{})=|t_2-t_1|

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If !,gamma is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of !,gamma at t_0 as

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:mbox{Speed}(t_0)=limsup_{t o t_0} {d(gamma(t),gamma(t_0))over |t-t_0|}

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and then

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:mbox{Length}(gamma)=int_a^b mbox{Speed}(t) , dt

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In particular, if X = mathbb{R}^n is Euclidean space and gamma : ightarrow mathbb{R}^n is differentiable then

Related Topics:
Euclidean space - Differentiable

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:mbox{Length}(gamma)=int_a^b left| , {dgamma over dt} , ight| , dt

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