Involute - or evolvent - curves are used as the shape of teeth of gears. For an introduction to more advanced math, see https://en.wikipedia.org/wiki/Involute.
The main reason for using gears with involute shapes is that there is no friction between teeth - against common believe. The teeth roll on each other when rotating.
It is the gear application that makes a polar calculation based on radii more useful than a Cartesian calculation based on (x,y) - we are only interested in the first small part that starts from the base circle.