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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.
nephroid: tangents as chords of a circle, principle nephroid: tangents as chords of a circle. Similar to the generation of a cardioid as envelope of a pencil of lines the following procedure holds: Draw a circle, divide its perimeter into equal spaced parts with points (see diagram) and number them consecutively.
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 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 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.
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 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.
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.