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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).
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. The classic Spirograph toy traces out epitrochoid and hypotrochoid curves.
The two angles , are defined differently (is one half of the rolling angle, is the parameter of the circle, whose chords are determined), for = one gets the same line. Hence any chord from the circle above is tangent to the nephroid and
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 generator circle is the inverse curve of the parabola's directrix.) This property gives rise to the following simple method to draw a cardioid: Choose a circle and a point on its perimeter, draw circles containing with centers on , and; draw the envelope of these circles.
A Roots blower is one extreme, a form of cycloid gear where the ratio of the pitch diameter to the generating circle diameter equals twice the number of lobes. In a two-lobed blower, the generating circle is one-fourth the diameter of the pitch circles, and the teeth form complete epi- and hypo-cycloidal arcs.
Bisect one of the angles made by these two lines and name the angle bisector b. Using a hyperbolic ruler, construct a line c such that c is perpendicular to b and parallel to a. As a result, c is also parallel to a', making c the common parallel to lines a and a'.