329 Parameterization

A function that labels points of a curve by a real parameter that is used to write coordinates along the curve.

An implicit equation gives a constraint on points; parameterization instead gives a rule that generates each point on the curve.

For example:

\[ x^2+y^2=1 \]

versus \[ x=\cos t,\qquad y=\sin t.\]

The first describes which points belong to the circle. The second tells you how to move through those points.

definition [d] (Parameterization = Parametrization = Parametric Representation) A continuous vector-valued function that traces a curve $C$ as the parameter varies over an interval $I$:

$\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle$ , $t \in I$ .

where

$\mathbf{r}(t)$ is the position vector of a point on $C$.
$C$ is the curve being traced.
$f, g, h$ are continuous real-valued functions on $I$.
$t$ is the parameter.
$I$ is the parameter interval.

Note:

a given curve admits many different parameterizations; speed and direction may differ.

definition [d] (Parameterization = Parametrization = Parametric Form) A description of a curve by expressing its coordinates as continuous functions of a single parameter $t \in I$:

$x = f(t)$, $y = g(t)$, $z = h(t)$ .

where

$f, g, h$ are continuous real-valued functions on $I$.
$t$ is the parameter.
$I$ is the parameter interval.
$x, y, z$ are coordinates of a point on the curve.

Note:

as $t$ increases, the point $(f(t), g(t), h(t))$ moves along the curve with a definite orientation.

329.1 Examples

329.1.1 Simple

A line in the plane labeled by three parameter values.

\[ \mathbf{r} : I \rightarrow \mathbb{R}^{2} \]

\[ I = \{ 0,\ 1,\ 2 \} \]

\[ \mathbf{r}(t) = \langle t,\ 2t \rangle \]

\[ \mathbf{r}(0) = \langle 0,\ 0 \rangle,\quad \mathbf{r}(1) = \langle 1,\ 2 \rangle,\quad \mathbf{r}(2) = \langle 2,\ 4 \rangle \]

where

$\mathbf{r}$ is the parameterization.
$I$ is the set of parameter values.
$t$ is the parameter.

329.1.2 General

A unit circle in the plane labeled by a continuous parameter, with sample points and a second parameterization of the same curve.

\[ \phi : [0, 2\pi) \rightarrow \mathbb{R}^{2} \]

\[ \phi(t) = \langle \cos t,\ \sin t \rangle \]

\[ x = \cos t,\quad y = \sin t \]

\[ \phi(0) = \langle 1,\ 0 \rangle,\quad \phi\!\left(\dfrac{\pi}{2}\right) = \langle 0,\ 1 \rangle,\quad \phi(\pi) = \langle -1,\ 0 \rangle \]

$Unit circle under \phi at t=0.$

Unit circle under $\phi$ at $t=0$.

$Unit circle under \phi at t=\pi/2.$

Unit circle under $\phi$ at $t=\pi/2$.

$Unit circle under \phi at t=\pi.$

Unit circle under $\phi$ at $t=\pi$.

The same circle admits another parameterization that traverses it twice as fast. Every point on the unit circle still has the form $\langle \cos\theta,\ \sin\theta \rangle$ for some angle $\theta$.

\[ \psi : [0, \pi) \rightarrow \mathbb{R}^{2} \]

To finish one full lap while $t$ runs only through $[0, \pi)$, the angle must advance twice as fast, so $\theta = 2t$. Substituting that angle into the same cosine and sine form gives

\[ \psi(t) = \langle \cos 2t,\ \sin 2t \rangle \]

so $\phi$ and $\psi$ share the same image while differing in speed. At the same parameter value $t=\pi/4$, the point $\psi(t)$ has already advanced farther around the circle than $\phi(t)$.

$Comparison of \phi and \psi at t=\pi/4.$

Comparison of $\phi$ and $\psi$ at $t=\pi/4$.

where

$\phi$ is a parameterization of the unit circle.
$\psi$ is another parameterization of the same circle.
$t$ is the parameter.
$\theta$ is the angle used inside cosine and sine.
$[0, 2\pi)$ and $[0, \pi)$ are the parameter intervals.
$x$ and $y$ are coordinates of a point on the curve.
the image of $\phi$ and of $\psi$ is the circle of radius $1$.

329.2 References

Stewart, J. Calculus. — parametrization via $\mathbf{r}(t)$ and parametric equations $x=f(t)$, $y=g(t)$, $z=h(t)$.
Bachman, D. A Geometric Approach to Differential Forms. Birkhäuser, 2012. — $\phi(t)=(\cos t,\sin t)$ for $0\leq t<2\pi$; also $\psi(t)=(\cos 2t,\sin 2t)$ for the same circle at twice the speed.

328 Notes

330 Partial Derivative