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Cyclogon Generated By A Star-like Quadrilateral
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon Generated By Triangle
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypocycloid
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. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line. Properties If the smaller circle has radius , and the larger circle has radius , then the parametric equations for the curve can be given by either: :\begin & x (\theta) = (R - r) \cos \theta + r \cos \left(\frac \theta \right) \\ & y (\theta) = (R - r) \sin \theta - r \sin \left( \frac \theta \right) \end or: :\begin & x (\theta) = r (k - 1) \cos \theta + r \cos \left( (k - 1) \theta \right) \\ & y (\theta) = r (k - 1) \sin \theta - r \sin \left( (k - 1) \theta \right) \end If is an integer, then the curve is closed, and has Cusp (singularity), cusps (i.e., sharp corners, where the curve is not Differentiable function, differentiable). Specially for the curve is a straight line and the circles are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epicycloid
In geometry, an epicycloid 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. It is a particular kind of roulette. Equations If the smaller circle has radius , and the larger circle has radius , then the parametric equations for the curve can be given by either: :\begin & x (\theta) = (R + r) \cos \theta \ - r \cos \left( \frac \theta \right) \\ & y (\theta) = (R + r) \sin \theta \ - r \sin \left( \frac \theta \right) \end or: :\begin & x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \\ & y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right) \end in a more concise and complex form :z(\theta) = r \left(e^ - (k + 1)e^ \right) where * angle is in turns: \theta \in , 2\pi * smaller circle has radius * the larger circle has radius Area (Assuming the initial point lies on the larger circle.) When is a pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). History The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians. Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity. English mathematician John Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon Generated By A Star-like Quadrilateral
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon Generated By A Convex Quadrilateral
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon Generated By A Non-convex Quadrilateral
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolate Cyclogon Generated By An Equilateral Triangle
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a ''prolate spheroid'', elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an ''oblate spheroid'', flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon Generated By A Square
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscribed Circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
