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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 b which rolls round a fixed circle of radius 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 a. The point P is at distance c from the centre of the circle of radius b. For this example a = 5, b = 7 and c = 2.2.

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

Hy`po*tro"choid (?), n. [Pref. hypo- + trochoid.] Geom.

A curve, traced by a point in the radius, or radius produced, of a circle which rolls upon the concave side of a fixed circle. See Hypocycloid, Epicycloid, and Trochoid.


© Webster 1913.

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