Parametric Cartesian equation:
x=(ab)cos(t)+ccos((a/b1)t),y=(ab)sin(t)csin((a/b1)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 hypotrochoid, an example of which is shown above, the circle of radius bb rolls on the inside of the circle of radius aa. The point PP is at distance cc from the centre of the circle of radius bb. For this examplea=5,b=7a = 5, b = 7and c = 2.22.2.

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

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Xah Lee