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Epicycloid
In geometry, an epicycloid (also called hypercycloid) 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. An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form. Equations If the rolling circle has radius r, and the fixed circle has radius R = kr, 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 This can be written in a more concise form using complex numbers as :z(\theta) = r \left( (k + 1)e^ - e^ \right) where * the angle \theta \in , 2\ ... [...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. History The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian people, Persian Islamic astronomy, astronomer and Islamic mathematics, mathematician Nasir al-Din al-Tusi in ''Tahrir al-Majisti (Commentary on the Almagest)''. German painter and German Renaissance theorist Albrecht Dürer described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively. Properties If the rolling circle has radius , and the fixed 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) \\ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deferent And Epicycle
In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. In particular it explained the apparent retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from the Earth. It was first proposed by Apollonius of Perga at the end of the 3rd century BC. It was developed by Apollonius of Perga and Hipparchus of Rhodes, who used it extensively, during the 2nd century BC, then formalized and extensively used by Ptolemy in his 2nd century AD astronomical treatise the '' Almagest''. Epicyclical motion is used in the Antikythera mechanism, itation requested/sup> an ancient Greek astronomical device, for compensating for the elliptical orbit of the Moon, moving faster at perigee and slower at apogee than cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle. The name was coined by Giovanni Salvemini in 1741 but the cardioid had been the subject of study decades beforehand.Yates Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the mic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Periodic Functions
This is a list of some well-known periodic functions. The constant function , where is independent of , is periodic with any period, but lacks a ''fundamental period''. A definition is given for some of the following functions, though each function may have many equivalent definitions. Smooth functions All trigonometric functions listed have period 2\pi, unless otherwise stated. For the following trigonometric functions: : is the th up/down number, : is the th Bernoulli number : in Jacobi elliptic functions, q=e^ Non-smooth functions The following functions have period p and take x as their argument. The symbol \lfloor n \rfloor is the floor function of n and \sgn is the sign function. K means Elliptic integral K(m) Vector-valued functions * Epitrochoid * Epicycloid (special case of the epitrochoid) * Limaçon (special case of the epitrochoid) * Hypotrochoid * Hypocycloid (special case of the hypotrochoid) * Spirograph (special case of the hypotrochoid) ... [...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 Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve generated by a curve rolling on another curve. The cycloid, with the Cusp (singularity), 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 Frequency, 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 field, 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-centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epitrochoid
In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric equations for an epitrochoid are: :\begin & x (\theta) = (R + r)\cos\theta - d\cos\left(\theta\right) \\ & y (\theta) = (R + r)\sin\theta - d\sin\left(\theta\right) \end The parameter is geometrically the polar angle of the center of the exterior circle. (However, is not the polar angle of the point (x(\theta),y(\theta)) on the epitrochoid.) Special cases include the limaçon with and the epicycloid with . The classic Spirograph toy traces out epitrochoid and hypotrochoid curves. The paths of planets in the once popular geocentric system of deferents and epicycles are epitrochoids with d>r, for both the outer planets and the inner planets. The orbit of the Moon, when centered around the Sun, approximates an epitrochoid. The com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclogon
In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that Rolling, rolls without slipping along a straight line. 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 that Circumscribed circle, circumscribes the polygon. For every cyclogon generated by a regular polygon, : A = P + 2C. \, Examples Cyclogons generated by an equilateral triangle and a square Prolate cyclogon generated by an equilateral triangle Curtate cyclogon generated by an equilateral triangle Cyclogons generated by quadrilaterals Generalized cyclogons A cyclogon is obtained when a polygon rolls over a straight line. Let it be assumed that the regular polygon rolls over the edge of another polygon. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epicyclic Gearing
An epicyclic gear train (also known as a planetary gearset) is a gear reduction assembly consisting of two gears mounted so that the center of one gear (the "planet") revolves around the center of the other (the "sun"). A carrier connects the centers of the two gears and rotates, to carry the planet gear(s) around the sun gear. The planet and sun gears mesh so that their pitch circles roll without slip. If the sun gear is held fixed, then a point on the pitch circle of the planet gear traces an epicycloid curve. An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of an outer gear ring, or ring gear, sometimes called an ''annulus gear''. Such an assembly of a planet engaging both a sun gear and a ring gear is called a planetary gear train.J. J. Uicker, G. R. Pennock and J. E. Shigley, 2003, ''Theory of Machines and Mechanisms,'' Oxford University Press, New York.B. Paul, 1979, ''Kinematics and Dynamics of Planar Machinery'', Pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epitrochoid
In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric equations for an epitrochoid are: :\begin & x (\theta) = (R + r)\cos\theta - d\cos\left(\theta\right) \\ & y (\theta) = (R + r)\sin\theta - d\sin\left(\theta\right) \end The parameter is geometrically the polar angle of the center of the exterior circle. (However, is not the polar angle of the point (x(\theta),y(\theta)) on the epitrochoid.) Special cases include the limaçon with and the epicycloid with . The classic Spirograph toy traces out epitrochoid and hypotrochoid curves. The paths of planets in the once popular geocentric system of deferents and epicycles are epitrochoids with d>r, for both the outer planets and the inner planets. The orbit of the Moon, when centered around the Sun, approximates an epitrochoid. The com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypotrochoid
In geometry, a hypotrochoid is a roulette (curve), roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The parametric equations for a hypotrochoid are: :\begin & x (\theta) = (R - r)\cos\theta + d\cos\left(\theta\right) \\ & y (\theta) = (R - r)\sin\theta - d\sin\left(\theta\right) \end where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from 0 to 2 \pi \times \tfrac (where is least common multiple). Special cases include the hypocycloid with and the ellipse with and . The eccentricity of the ellipse is :e=\frac becoming 1 when d=r (see Tusi couple). The classic Spirograph toy traces out hypotrochoid and epitrochoid curves. Hypotrochoids describe the support of the eigenvalues of some random matrices with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. On a basic level, it is the path traced by a curve while rolling on another curve without slipping. Definition Informal definition Roughly speaking, a roulette is the curve described by a point (called the ''generator'' or ''pole'') attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve, in the fixed plane called a roulette. Special cases and related concepts In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
