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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 curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
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.
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 3 N {\displaystyle 3N} points (see diagram) and number them consecutively.
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.
Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids.The well-known toy version was developed by British engineer Denys Fisher and first sold in 1965.
Suppose X and Y in the Lie algebra are given. Their exponentials, exp(X) and exp(Y), are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some Z in the Lie algebra, exp(Z) = exp(X) exp(Y), and one may tentatively write = (,), for C some expression in X and Y.