Parametrisation of curves in the plane

The exponential form $z(\theta)=\rho\,e^{i\theta}$ is already a parametrisation: as $\theta\in[0,2\pi)$ varies, $z(\theta)$ describes the circle of radius $\rho$. It is a kinematic point of view: the curve is the image of the motion of a point as a parameter varies. Let us look at three examples.

Circle. Centre $(x_0,y_0)$, radius $r$: $\begin{cases} x(t) = x_0 + r\cos t \\ y(t) = y_0 + r\sin t\end{cases}, \quad t\in[0,2\pi).$ Check: $(x-x_0)^2 + (y-y_0)^2 = r^2(\cos^2 t + \sin^2 t) = r^2$. The parametrisation traverses the circle anticlockwise exactly once.

Ellipse. Canonical form $x^2/a^2 + y^2/b^2 = 1$: $\begin{cases} x(t) = a\cos t \\ y(t) = b\sin t\end{cases}, \quad t\in[0,2\pi).$ Note: $t$ is not the geometric angle of the point with respect to the centre (unless $a=b$). It is an auxiliary angular parameter: in the astronomical formalism of orbits it is called the eccentric anomaly, following the Keplerian terminology taken up by Stillwell.

Lissajous curve. Two perpendicular oscillations of different frequencies: $\begin{cases} x(t) = A\sin(\omega_1 t) \\ y(t) = B\sin(\omega_2 t + \delta)\end{cases}.$ If $\omega_1/\omega_2$ is rational, the curve is closed; if irrational, it densely fills the rectangle $[-A,A]\times[-B,B]$. The case $\omega_2/\omega_1 = 1{:}1$ with $\delta=\pi/2$ gives a circle ($A=B$) or an ellipse with parallel axes; the ratio $1{:}2$ gives the classic figure of eight.

Links

Topics: Complex numbers
Concepts: Eccentric anomaly · Circle · Lissajous curve · Ellipse · Exponential form · Parametrisation
Skills: Model
People: Johannes Kepler (Keplero) · Lissajous