Regular curve

Definition

Let $I \subseteq \mathbb{R}$ be an interval, and let $\gamma \colon I \to \mathbb{R}^n$ be a $C^1$ parametrised curve. The curve $\gamma$ is regular if its tangent vector $\gamma'(t)$ is not zero for each $t \in I$.

Remarks

The circle $t \mapsto (\cos t, \sin t)$ and each line $t \mapsto p + t v$ with $v \ne 0$ are regular. The curve $t \mapsto (t^2, t^3)$ is not regular, since its tangent vector is zero at $t = 0$. The curve $t \mapsto (t^3, t^3)$ is not regular at $t = 0$, but its image is a straight line. So regularity is a property of the parametrisation, not of the image. A regular curve has an arc length parametrisation.

Explanations

L1 human

A regular curve never stops. The moving point always has non-zero speed, so the curve has no corners or cusps caused by the point coming to rest.