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| [[image:CycloidAnim04.gif|thumb|290px|right|A [[cycloid]] (a common trochoid) generated by a rolling circle]]
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| [[File:Cycloids.svg|thumb|290px|right|From top to bottom a prolate, common and curtate cycloid respectively, with ''b'' being the same for all curves and with λ = ''a'' / ''b'']]
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| '''Trochoid''' derives from the Greek word for wheel, "trokhos". In these sense, the word "trochoid" was coined by [[Gilles de Roberval]] for the [[curve]] described by a fixed point as a [[circle]] rolls along a straight line. As a circle of radius ''a'' rolls without slipping along a line ''L,'' the center ''C'' moves parallel to ''L,'' and every other point ''P'' in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let ''CP = b''. If ''P'' lies inside the circle (''b'' < ''a''), on its circumference (''b'' = ''a''), or outside (''b'' > ''a''), the trochoid is described as being curtate, common, or prolate, respectively. [[Parametric equation]]s of the trochoid, which assume ''L'' is the x-axis, are
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| :<math>x = a\theta - b \sin(\theta) \,</math>
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| :<math>y = a - b \cos(\theta) \,</math>
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| where ''θ'' is the variable angle through which the circle rolls. A curtate trochoid is traced by a pedal when a bicycle is pedaled along a straight line. A [[prolate]], or extended trochoid is traced by the tip of a paddle when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a [[cycloid]], has [[cusp (singularity)|cusp]]s at the points where ''P'' touches the ''L''.
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| A more general approach would define a trochoid as the [[Locus (mathematics)|locus]] of a point <math>(x,y)</math> [[orbit]]ing at a constant rate around an axis located at <math>(x',y')</math>,
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| :<math>x=x'+r_1\cos(\omega_1 t+\phi_1),\ y=y'+r_1\sin(\omega_1 t+\phi_1),\ r_1>0,</math>
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| which axis is being translated in the ''x-y''-plane at a constant rate in ''either'' a straight line,
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| :<math>\begin{array}{lcl}
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| x'=x_0+v_{2x} t,\ y'=y_0+v_{2y} t\\
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| \therefore x = x_0+r_1\cos(\omega_1 t+\phi_1)+v_{2x} t,\ y = y_0+r_1 \sin(\omega_1 t+\phi_1)+v_{2y} t,\\
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| \end{array}</math>
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| or a circular path (another orbit) around <math>(x_0,y_0)</math> (the [[hypotrochoid]]/[[epitrochoid]] case),
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| :<math>\begin{array}{lcl}
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| x' = x_0+r_2\cos(\omega_2 t+\phi_2),\ y' = y_0+r_2\sin(\omega_2 t+\phi_2),\ r_2\ge 0\\
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| \therefore x = x_0+r_1\cos(\omega_1 t+\phi_1)+r_2\cos(\omega_2 t+\phi_2),\ y = y_0+r_1 \sin(\omega_1 t+\phi_1)+r_2\sin(\omega_2 t+\phi_2),\\
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| \end{array}</math>
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| The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a [[Periodic function|periodic]] (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, <math>\omega_1/\omega_2</math>, is a rational number, say <math>p/q</math>, where <math>p</math> & <math>q</math> are [[coprime]], in which case, one period consists of <math>p</math> orbits around the moving axis and <math>q</math> orbits of the moving axis around the point <math>(x_0,y_0)</math>. The special cases of the [[epicycloid]] and [[hypocycloid]], generated by tracing the locus of a point on the perimeter of a circle of radius <math>r_1</math> while it is rolled on the perimeter of a stationary circle of radius <math>R</math>, have the following properties:
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| :<math>\begin{array}{lcl}
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| \text{epicycloid: }&\omega_1/\omega_2&=p/q=r_2/r_1=R/r_1+1,\ |p-q| \text{ cusps}\\
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| \text{hypocycloid: }&\omega_1/\omega_2&=p/q=-r_2/r_1=-(R/r_1-1),\ |p-q|=|p|+|q| \text{ cusps}
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| \end{array}</math>
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| where <math>r_2</math> is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima."
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| ==See also==
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| * [[List of periodic functions]]
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| * [[Epitrochoid]]
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| * [[Hypotrochoid]]
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| * [[Cycloid]]
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| * [[Epicycloid]]
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| * [[Hypocycloid]]
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| * [[Spirograph]]
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| ==External links==
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| * http://www.xahlee.org/SpecialPlaneCurves_dir/Trochoid_dir/trochoid.html
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| * {{MathWorld | urlname=Trochoid | title=Trochoid}}
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Trochoid Online experiments with the Trochoid using JSXGraph]
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| [[Category:Curves]]
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Greetings! I am Marvella and I really feel comfy when individuals use the full title. California is our birth place. Managing individuals is his profession. To gather cash is 1 of the things I adore most.
Here is my page - at home std testing