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Mathematics · 术语表

什么是 Change of parameter？

又称： geometric arc

Definition 18.2 University Mathematics — Year 2 · 第 18 章 — Curves

A change of parameter of class $\mathcal{C}^k$ is a $\mathcal{C}^k$ diffeomorphism $\theta \colon J \to I$ between intervals ( $\theta' \neq 0$ everywhere). The arcs $\gamma$ and $\gamma \circ \theta$ are called equivalent; a geometric arc (or curve) is an equivalence class. Notions invariant under change of parameter — the trajectory, the tangent line, arc length, curvature — are called geometric.

示例

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.

Example 18.5

The arc $\gamma(t) = (t^2, t^3)$ is $\mathcal{C}^\infty$ but not regular: $\gamma'(0) = (0, 0)$ . Its trajectory, the semicubical parabola $y^2 = x^3$ , has a cusp at the origin: smoothness of the parametrization does not prevent a geometric singularity where the velocity vanishes. This is why the regularity hypothesis $\gamma' \neq 0$ is not cosmetic.

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