Centered trochoid

In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle.

Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle. Specifically, a centered trochoid is a curve that can be parameterized in the complex plane by

or in the Cartesian plane by

where

If ${\displaystyle \omega _{1}/\omega _{2}}$ is rational then the curve is closed and algebraic. Otherwise the curve winds around the origin an infinite number of times, and is dense in the annulus with outer radius ${\displaystyle |r_{1}|+|r_{2}|}$ and inner radius ${\displaystyle ||r_{1}|-|r_{2}||}$.

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