Parametric Cartesian equation:
x=(a+b)cos(t)ccos((a/b+1)t),y=(a+b)sin(t)csin((a/b+1)t)x = (a + b) \cos(t) - c \cos((a/b + 1)t), y = (a + b) \sin(t) - c \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 epitrochoid, 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 at distance cc from the centre of the circle of radius bb. For the example a=5,b=3a = 5, b = 3 and c=5c = 5 (so PP goes inside the circle of radius aa).

An example of an epitrochoid appears in Dürer's work Instruction in measurement with compasses and straight edge(1525). He called them spider lines because the lines he used to construct the curves looked like a spider.

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

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Xah Lee