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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.
In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be slightly over 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.
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 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).
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 ...
The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 4).
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.