**Parametric Cartesian equation: **

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

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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

For the epicycloid, an example of which is shown above, the circle of radius *b* rolls on the outside of the circle of radius *a*. The point *P* is on the circumference of the circle of radius *b*. For the example drawn here *a* = 8 and *b* = 5.

These curves were studied by Dürer (1525), Desargues (1640), Huygens (1679), Leibniz, Newton (1686), de L'Hôpital (1690), Jacob Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Daniel Bernoulli (1725), Euler (1745, 1781).

Special cases are *a* = *b* when a cardioid is obtained and *a* = 2*b* when a nephroid is obtained.

If *a* = (*m* - 1)*b* where *m* is an integer, then the length of the epicycloid is 8*mb* and its area is π*b*^{2}(*m*^{2} + *m*).

The pedal curve, when the pedal point is the centre, is a rhodonea curve.

The evolute of an epicycloid is a similar epicycloid - look at the evolute of the epicycloid above to see it is a similar epicycloid but smaller in size.

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JOC/EFR/BS January 1997

The URL of this page is:

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