Visualising maths

Parametric equations

We often define a curve by expression $x$ as a function of $y$:

$$y = f(x)$$

Using parametric equations we define the $x$ and $y$ coordinates of the points on the curve in terms of an independent variable, which we will call $t$:

$$ \begin{align} x = g(t)\\ y = h(t) \end{align} $$

For any value of $t$, a value of $x$ and $y$ can be calculated, and the point $(x, y)$ will lie on the curve.

One way to think of this is to imagine $t$ representing time. As the time changes, the point $(x, y)$ will move, tracing the curve. But this is just an aid to understanding, the parameter $t$ does not necessarily represent time.

In this section we will look at the parametric equations of parabolas and hyperbolas, and also see how to express them as Cartesian equations.