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The path traced out by a point on the circumference of a circle of radius b rolling on the outside of a circle of radius a. It is described by the parametric equations:

      x = (a + b) cos (t) - b cos [(a/b + 1)t ],
      y = (a + b) sin (t) - b sin [(a/b + 1)t ].

An epicycloid is like a cycloid on the circumference of a circle and is closely related to the epitrochoid, hypocycloid, and hypotrochoid.

An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid (after the buttercup genus Ranunculus).

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