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 tt) and then xx and yy (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 xx and yy.

x2+y2=1x^{2}+y^{2}=1 (4.115)

But we could also write it as two separate equations - one for xx in terms of a new variable we’ll introduce, tt, and one for yy in terms of tt.

x=cos(t)\displaystyle x=\cos(t) (4.116)
y=sin(t)\displaystyle y=\sin(t) (4.117)

We can get from the parametric equations (the ones in terms of tt) to the Cartesian equations using a little algebra. Adding together x2x^{2} and y2y^{2} gives this equation.

x2+y2=cos2(t)+sin2(t)x^{2}+y^{2}=\cos^{2}(t)+\sin^{2}(t) (4.118)

As cos2(t)+sin2(t)=1\cos^{2}(t)+\sin^{2}(t)=1, the overall result is that

x2+y2=1x^{2}+y^{2}=1 (4.119)

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