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**Parametric Cartesian equation: x = ( a - b)cos(t) + ccos((a/b -1)t), y = (a - b)sin(t) - csin((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 * P* on a circle of radius

*which rolls round a fixed circle of radius*

**b***.*

**a**
For the hypotrochoid, shown above in hit or miss ASCII, the circle of radius * b* rolls on the inside of the circle of radius

*. The point*

**a***is at distance*

**P***from the centre of the circle of radius*

**c***. For this example*

**b***= 5,*

**a***= 7 and*

**b***= 2.2.*

**c**These curves were studied by Newton, la Hire, Desargues, and Leibniz amoung others.