Epicycloid

In geometry, an epicycloid(also called hypercycloid) is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette. An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form. If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either: or: in a more concise and complex form where angle θ is in turns: smaller circle has radius r the larger circle has radius kr (Assuming the initial point lies on the larger circle.) When k is a positive integer, the area of this epicycloid is It means that the epicycloid is larger than the original stationary circle. If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners). If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has p cusps. Count the animation rotations to see p and q If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r. The distance from (x = 0, y = 0) origin to (the point p on the small circle) varies up and down as where R = radius of large circle and 2r = diameter of small circle File:Epicycloid-1.svg| {{math|1=''k'' = 1 }}; a ''[[cardioid]]'' File:Epicycloid-2.svg| {{math|1=''k'' = 2 }}; a ''[[nephroid]]'' File:Epicycloid-3.svg| {{math|1=''k'' = 3 }}; a ''trefoiloid'' File:Epicycloid-4.svg| {{math|1=''k'' = 4 }}; a ''quatrefoiloid'' File:Epicycloid-2-1.svg| {{math|1=''k'' = 2.1 = 21/10}} File:Epicycloid-3-8.svg| {{math|1=''k'' = 3.8 = 19/5}} File:Epicycloid-5-5.svg| {{math|1=''k'' = 5.5 = 11/2}} File:Epicycloid-7-2.svg| {{math|1=''k'' = 7.2 = 36/5}} The epicycloid is a special kind of epitrochoid. An epicycle with one cusp is a cardioid, two cusps is a nephroid. An epicycloid and its evolute are similar.