---
tag: 0JF1XXGWB4
kind: definition
name: Parametrised curve
msc: [53A04]
defines: 0G7WM8DX4P
uses: [018GTRS7A4, 0S9EK5MQPE, 0TCBZMEYGN]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiable_curve
    locator: Definitions
  - url: https://en.wikipedia.org/wiki/Parametric_equation
    locator: Parametric plane curves
---

## Definition

Let $I \subseteq \mathbb{R}$ be an interval with more than one point.
A parametrised curve in $\mathbb{R}^n$ is a [[continuous|0S9EK5MQPE]] 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.