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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 which are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a.

For the epitrochoid, an example of which is shown above, the circle of radius b rolls on the outside of the circle of radius a. The point P is at distance c from the centre of the circle of radius b. For the example a = 5, b = 3 and c = 5 (so P is actually inside the circle of radius a).

An example of an epitrochoid appears in Durer's work "Instruction in Measurement With Compasses and Straight Edge". He called them "spider lines" due to the graph looking (slightly, I guess) like a spider.

Ep`i*tro"choid (?), n. [Pref. epi- + Gr. wheel + -oid.] Geom.

A kind of curve. See Epicycloid, any Trochoid.

 

© Webster 1913.

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