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line kurve

Definition

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

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

Properties

Special points

Bogenlänge

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

If n=2n = 2 (2D): L(γ)=abẋ2+ẏ2dtL(\gamma) = \int_a^b \sqrt{\dot x^2 + \dot y^2}\,\text{d}t