**Parametric Cartesian equation: **

*x* = (*a* - *b*) cos(*t*) + *c* cos((*a*/*b* -1)*t*), *y* = (*a* - *b*) sin(*t*) - *c* sin((*a*/*b* -1)*t*)

**Click below to see one of the Associated curves.**

If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a.

For the hypotrochoid, an example of which is shown above, the circle of radius *b* rolls on the inside of the circle of radius *a*. The point *P* is at distance *c* from the centre of the circle of radius *b*. For this example *a* = 5, *b* = 7 and c = 2.2.

These curves were studied by la Hire, Desargues, Leibniz, Newton and many others.

**Other Web site:**

JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Hypotrochoid.html