In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around without slipping around a fixed circle. It is a particular kind of roulette. Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ... In the differential geometry of curves, a roulette is the general concept behind cycloids, epicycloids, hypocycloids, and involutes. ...

An epicycloid with n − 1 cusps is given by the parametric equations Image File history File links Epicycloid. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

The epicycloid is a special kind of epitrochoid. An epitrochoid is a roulette traced by a point attached to a circle of radius b rolling around the outside of a fixed circle of radius a, where the point is a distance h from the center of the exterior circle. ...

An epicycle with one cusp is a cardioid. In geometry, the cardioid is an epicycloid which has one and only one cusp. ...

An epicycloid and its evolute are similar.[1] In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... Several equivalence relations in mathematics are called similarity. ...

See also: cycloid, hypocycloid, deferent and epicycle. Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... In geometry, a hypocycloid is a special plane curve, a roulette, generated by the trace of a fixed point on a small circle that rolls within a larger circle. ... In the Ptolemaic system of astronomy, the epicycle (literally: on the cycle in Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ...