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line kurve

Definition

A curve is a generalization of a line, in that its curvature need not be zero.

$${\boldsymbol \gamma}:[a,b] \rightarrow {\mathop{\mathbb R}}^n, t \mapsto {\begin{pmatrix} \gamma_1(t) \\ \vdots \\ \gamma_n(t) \end{pmatrix}}$$

Properties

Special points

Bogenlänge

Die Bogenlänge einer Kurve $L(\gamma)$ ist $$L(\gamma) = \int_{a}^{b} {\left\lVert{\dot \gamma(t)}\right\rVert} {\,\text{d}}t$$

If $n = 2$ (2D): $L(\gamma) = \int_a^b \sqrt{\dot x^2 + \dot y^2}{\,\text{d}}t$

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