Epitrochoid

Parametric Cartesian equation:
x = (a + b) cos(t) - c cos((a/b + 1)t), y = (a + b) sin(t) - c sin((a/b + 1)t)


Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

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 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 goes inside the circle of radius a).

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.

Other Web site:

Xah Lee

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JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Epitrochoid.html