Whereas the cycloid is generated by a point on the rim of a wheel rolling along a straight line, a related type of curve arises from a wheel rolling on the outside of a second, fixed wheel. The resulting curve is an epicycloid (from the Greek epi, meaning "over" or "above"). Alternatively we can let the wheel roll along the inside of a fixed wheel, generating a hypocycloid (hypo = "under"). The epicycloid and hypocycloid come in a great variety of shapes, depending on the ratio of the radii of the two wheels. Let the radii of the fixed and moving wheels be R and r, respectively. If R/r is a fraction in lowest terms, say m, the curve will have m cusps (corners), and it will be completely traced after n full rotations around the fixed wheel.
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