What is Parametrized arc? — Mathematics

Definition 18.1 University Mathematics — Year 2 · Chapter 18 — Curves

A parametrized arc of class $\mathcal{C}^k$ ( $k \geq 1$ ) is a map $\gamma \colon I \to \R^n$ of class $\mathcal{C}^k$ on an interval $I$ . A point $\gamma(t)$ is regular if $\gamma'(t) \neq 0$ , and the arc is regular if all its points are. The line through $\gamma(t)$ directed by $\gamma'(t)$ is the tangent line at a regular point.

Examples

Example 18.4 (Speed changes nothing geometric)

Parametrize the unit circle by

\gamma(t) = (\cos t^2,\ \sin t^2), \qquad t \in \intcc0{\sqrt{2\pi}} .

The speed $\norm{\gamma'(t)} = 2t$ grows linearly, yet

L = \int_0^{\sqrt{2\pi}}2t\,\dd t = 2\pi ,

the same length as at constant speed — as Theorem 18.7 promises, via the change of parameter $t \mapsto t^2$ . The tangent line, the curvature computed from Proposition 18.17, and every other geometric quantity agree as well; only $t = 0$ deserves a glance, where $\gamma'(0) = 0$ makes this parametrization irregular although the trajectory is a perfect circle. Geometric statements tolerate bad parametrizations badly: reparametrize first, conclude second.

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