Change of parameter কী?

Example 18.4 (Speed changes nothing geometric)

Parametrize the unit circle by

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

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.