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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).
Let the parameter be the angle by which the tangent point rotates on , and ′ be the angle by which rotates (i.e. by which travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by B {\displaystyle B} and T {\displaystyle T} along their respective circles must be the same, therefore
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 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.
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 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.
Three distinct points create a unique circle [4] Given any two lines, they meet at a unique point [4] (normally, this would contradict the parallel axiom of hyperbolic geometry, since there can be many different lines parallel to the same line [1]) Angle measures have signs. Here, they will be defined in the following way: Consider a triangle XYZ.