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|
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.
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