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In geometry, an epicycloid (also called hypercycloid) [1] 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.
The cycloid through the origin, generated by a circle of radius r rolling over the x-axis on the positive side (y ≥ 0), consists of the points (x, y), with = () = (), where t is a real parameter corresponding to the angle through which the rolling circle has rotated. For given t, the circle's centre lies at (x, y) = (rt, r).
If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, the point lies on the circle then the roulette is a cycloid. A related concept is a glissette, the curve described by a point attached to a given curve as it slides along two (or more) given curves.
The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving an astroid). In 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.
The epitrochoid with R = 3, r = 1 and d = 1/2. In geometry, an epitrochoid (/ ɛ p ɪ ˈ t r ɒ k ɔɪ d / or / ɛ p ɪ ˈ t r oʊ k ɔɪ d /) 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 cyclocycloid (in this case an epicycloid) with R = 3, r = 1 and d = 1/2. A cyclocycloid is a roulette traced by a point attached to a circle of radius r rolling around, a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.
A cycloid (as used for the flank shape of a cycloidal gear) is constructed by rolling a rolling circle on a base circle. If the diameter of this rolling circle is chosen to be infinitely large, a straight line is obtained. The resulting cycloid is then called an involute and the gear is called an involute gear. In this respect involute gears ...
The hypocycloid traced by any point on the pitch circle of the smaller gear is a diameter of the larger gear. The mechanism has been used in Murray's Hypocyclic Engine. Trammel of Archimedes. Originally an ellipsograph. As a mechanism, it uses the fact that a circle and a straight line are special cases of an ellipse.