4.10 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 ) and then and (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 and .
But we could also write it as two separate equations - one for in terms of a new variable we’ll introduce, , and one for in terms of .
We can get from the parametric equations (the ones in terms of ) to the Cartesian equations using a little algebra. Adding together and gives this equation.
As , the overall result is that
which is the Cartesian equation of a circle with modulus one!