HypotrochoidIn geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. The parametric equations for a hypotrochoid are: where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle). When measured in radian, θ takes values from 0 to (where LCM is least common multiple).
EpicycloidIn 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.
HypocycloidIn geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line. The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in Tahrir al-Majisti (Commentary on the Almagest).
EpitrochoidIn geometry, an epitrochoid (ɛpᵻˈtrɒkɔɪd or ɛpᵻˈtroʊkɔɪd) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle. The parametric equations for an epitrochoid are The parameter θ is geometrically the polar angle of the center of the exterior circle. (However, θ is not the polar angle of the point on the epitrochoid.) Special cases include the limaçon with R = r and the epicycloid with d = r.
CycloidIn geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve).
