A calculus of the absurd

4.9 Parametric equations

If you’ve ever been to a science museum, then you may have seen a kind of device where if you turn a handle connected to a cog, that cog spins a bunch of other cogs. Although all the cogs spin at different rates, they’re all ultimately driven by the cog which you’re spinning.

This is a bit like how parametric equations work - we create a "parameter" (the cog which you spin, and often named \(t\)) and then \(x\) and \(y\) (or whatever the axes are called) are a bunch of other cogs connected to the initial cog.

For example, we can write the equation of the unit circle in terms of \(x\) and \(y\).

\begin{equation} x^2 + y^2 = 1 \end{equation}

But we could also write it as two separate equations - one for \(x\) in terms of a new variable we’ll introduce, \(t\), and one for \(y\) in terms of \(t\).

\begin{align} & x = \cos (t) \\ & y = \sin (t) \end{align}

We can get from the parametric equations (the ones in terms of \(t\)) to the Cartesian equations using a little algebra. Adding together \(x^2\) and \(y^2\) gives this equation.

\begin{equation} x^2 + y^2 = \cos ^2 (t) + \sin ^2 (t) \end{equation}

As \(\cos ^2 (t) + \sin ^2 (t) = 1\), the overall result is that

\begin{equation} x^2 + y^2 = 1 \end{equation}

which is the Cartesian equation of a circle with modulus one!