From Osher Doctorow Ph.D.

To summarize the relationships between/among cardioids, cycloids, epicycloids, and hypocycloids, after correcting my confusion of cardioid and cycloid, here is the story. All of these are curves generated by rolling circles. While cycloids are generated only by one rolling circle (being the slowly oscillating path traced out by a point on the circumference of a circle that rolls without slipping along a straight line in one direction, cardioids are special cases of epicycloids. Epicycloids are similarly generated by one circle rolling over a second fixed circle on the outside of the latter - the point of tangency (touching) traces out an epicycloid, and for epicycloids the rolling circle is smaller than or equal to the other circle in radius. When the radii are equal, it's called a cardioid. The hypocycloid is like the epicycloid but with the smaller circle rolling inside the fixed larger one. The cardioid has one cusp, the nephroid is like the cardioid except that it has 3 equally spaced cusps, and the ranunculoid is like the others but has 5 equally spaced cusps. The cycloid, epicycloid, and hypocycloid have the following respective equations: 1) x = a(t - sin(t)) y = a(1 - cos(t)) 2) x = (a + b)cos(t) - bcos[(a + b)t/b] y = (a + b)sin(t) - bsin[(a + b)t/b] 3) x = (a - b)cos(t) + bcos[(a - b)t/b] y = (a - b)sin(t) - bsin[(a - b)t/b] The cardioid is the second case when a = b, which results in: 4) x = 2acos(t) - acos(2t) y = 2asin(t) - asin(2t) Osher Doctorow Ph.D.