Parametrised curve

Definition

Let $I \subseteq \mathbb{R}$ be an interval with more than one point. A parametrised curve in $\mathbb{R}^n$ is a continuous map $\gamma \colon I \to \mathbb{R}^n$, $\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t))$. Its image, or trace, is the set $\gamma(I) = \{\gamma(t) : t \in I\}$. For $r \ge 1$, the curve is a $C^r$ curve if each component $\gamma_i$ is $r$ times continuously differentiable on $I$. When $I = [a, b]$, the points $\gamma(a)$ and $\gamma(b)$ are the initial point and the end point of $\gamma$.

Remarks

A curve is different from its image. The maps $t \mapsto (\cos t, \sin t)$ on $[0, 2\pi]$ and $t \mapsto (\cos 2t, \sin 2t)$ on $[0, 2\pi]$ have the same image, the unit circle, but the second goes around it twice. The helix $\gamma(t) = (\cos t, \sin t, t)$, $t \in \mathbb{R}$, is a $C^\infty$ curve in $\mathbb{R}^3$. The curve is simple if $\gamma$ is injective, and it is closed if $I = [a, b]$ and $\gamma(a) = \gamma(b)$.

Explanations

L1 human

A parametrised curve tells where a moving point is at each time $t$. It records both the path and the way the point travels along it, including its speed and direction.