# What are parametric equations used for?

Oct 5, 2014

Parametric equations are useful when a position of an object is described in terms of time $t$. Let us look at a couple of example.

Example 1 (2-D)

If a particle moves along a circular path of radius r centered at $\left({x}_{0} , {y}_{0}\right)$, then its position at time $t$ can be described by parametric equations like:

$\left\{\begin{matrix}x \left(t\right) = {x}_{0} + r \cos t \\ y \left(t\right) = {y}_{0} + r \sin t\end{matrix}\right.$

Example 2 (3-D)

If a particle rises along a spiral path of radius r centered along the $z$-axis, then its position at time $t$ can be described by parametric equations like:

$\left\{\begin{matrix}x \left(t\right) = r \cos t \\ y \left(t\right) = r \sin t \\ z \left(t\right) = t\end{matrix}\right.$

Parametric equations are useful in these examples since they allow us to describe each coordinate of the position of a particle separately in terms of time.

I hope that this was helpful.