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The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).. 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.
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).
Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads: Construct an angle equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools: an unmarked straightedge, and; a compass.
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.
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.
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 ...
A circle (C 3) centered at B' with radius |B'B| meets the circle (C 2) at A'. A circle (C 4) centered at A' with radius |A'A| meets the circle (C 1) at E and E'. Two circles (C 5) centered at E and (C 6) centered at E' with radius |EA| meet at A and O. O is the sought center of |AD|. The design principle can also be applied to a line segment AD.