Parametric Cartesian equation:
x=(a+b)cos(t)bcos((a/b+1)t),y=(a+b)sin(t)bsin((a/b+1)t)x = (a + b) \cos(t) - b \cos((a/b + 1)t), y = (a + b) \sin(t) - b \sin((a/b + 1)t)


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 PP on a circle of radius bb which rolls round a fixed circle of radius aa.

For the epicycloid, an example of which is shown above, the circle of radius bb rolls on the outside of the circle of radius aa. The point PP is on the circumference of the circle of radius bb. For the example drawn here a=8a = 8 and b=5b = 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=ba = b when a cardioid is obtained and a=2ba = 2b when a nephroid is obtained.

If a=(m1)ba = (m - 1)b where mm is an integer, then the length of the epicycloid is 8mb8mb and its area is πb2(m2+m)\pi 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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Xah Lee