**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.**

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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.

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JOC/EFR/BS January 1997

The URL of this page is:

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