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Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. The evolute of an involute is the original curve. It is generalized by the Roulette (curve), roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled ''Horologium Oscillatorium, Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing the cycloidal pendulum, which has the useful property that its period is independent of the amplitude of oscillation. Involute of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolute
In the differential geometry of curves, the evolute of a curve is the locus (mathematics), locus of all its Center of curvature, centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope (mathematics), envelope of the perpendicular, normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic (mathematics), caustic of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes. History Apollonius of Perga, Apollonius ( 200 BC) discussed evolut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycloidal Pendulum
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). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid. History The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
