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| In the [[differential geometry of curves]], a '''roulette''' is a kind of [[curve]], generalizing [[cycloid]]s, [[epicycloid]]s, [[hypocycloid]]s, [[trochoid]]s, and [[involute]]s.
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| == Definition ==
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| === Informal definition ===
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| [[Image:RouletteAnim.gif||right|frame|Construction of a roulette: specifically, a [[cissoid of Diocles]].]]
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| Roughly speaking, a roulette is the curve described by a point (called the ''generator'' or ''pole'') attached to a given curve as it 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.
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| In the illustration, the fixed curve (blue) is a [[parabola]], the rolling curve (green) is an equal parabola, and the generator is the vertex of the rolling parabola which describes the roulette (red). In this case the roulette is the [[cissoid of Diocles]].<ref name="2dcurves_cubicc"/>
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| === Special cases and related concepts ===
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| In the case where the rolling curve is a [[line (geometry)|line]] and the generator is a point on the line, the roulette is called an [[involute]] of the fixed curve. 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]].
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| 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.
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| === Formal definition === | |
| Formally speaking, the curves must be [[differentiable]] curves in the [[Euclidean plane]]. The ''fixed curve'' is kept invariant; the ''rolling curve'' is subjected to a [[continuous function|continuous]] [[congruence (geometry)|congruence]] transformation such that at all times the curves are [[tangent]] at a point of contact that moves with the same speed when taken along either curve (another way to express this constraint is that the point of contact of the two curves is the [[instant centre of rotation]] of the congruence transformation). The resulting roulette is formed by the [[locus (mathematics)|locus]] of the generator subjected to the same set of congruence transformations.
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| Modeling the original curves as curves in the [[complex plane]], let <math>r,f:\mathbb R\to\mathbb C</math> be the two [[Differential geometry of curves#Length_and_natural_parametrization|natural parameterizations]] of the rolling (<math>r</math>) and fixed (<math>f</math>) curves, such that <math>r(0)=f(0)</math>, <math>r^\prime(0)=f^\prime(0)</math>, and <math>|r^\prime(t)|=|f^\prime(t)| \neq 0</math> for all <math>t</math>. The roulette of generator <math>p\in\mathbb C</math> as <math>r</math> is rolled on <math>f</math> is then given by the mapping:
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| :<math>t\mapsto f(t)+(p-r(t)){f'(t)\over r'(t)}.</math>
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| == Generalizations ==
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| If, instead of a single point being attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent curves is produced. The envelope of this family may also be called a roulette.
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| Roulettes in higher spaces can certainly be imagined but one needs to align more than just the tangents.
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| ==Example==
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| If the fixed curve is a [[catenary]] and the rolling curve is a [[line (mathematics)|line]], we have:
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| :<math>f(t)=t+i(\cosh(t)-1) \qquad r(t)=\sinh(t)</math>
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| :<math>f'(t)=1+i\sinh(t) \qquad r'(t)=\cosh(t).</math>
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| The parameterization of the line is chosen so that
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| :<math>|f'(t)| \,</math> <math>=\sqrt{1^2+\sinh^2(t)}</math> <math>=\sqrt{\cosh^2(t)}</math> <math>=|r'(t)|. \,</math>
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| Applying the formula above we obtain:
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| :<math>f(t)+(p-r(t)){f'(t)\over r'(t)}
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| =t-i+{p-\sinh(t)+i(1+p\sinh(t))\over\cosh(t)}
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| =t-i+(p+i){1+i\sinh(t)\over\cosh(t)}.</math>
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| If ''p'' = −''i'' the expression has a constant imaginary part (namely −''i'') and the roulette is a horizontal line. An interesting application of this is that a [[square wheel]] could roll without bouncing on a road that is a matched series of catenary arcs.
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| ==List of roulettes==
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| {| class="wikitable"
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| |-
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| ! Fixed curve
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| ! Rolling curve
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| ! Generating point
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| ! Roulette
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| |-
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| | ''Any curve''
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| | [[Line (mathematics)|Line]]
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| | Point on the line
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| | [[Involute]] of the curve
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Circle]]
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| | ''Any''
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| | [[Trochoid]]
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Circle]]
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| | Point on the circle
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| | [[Cycloid]]
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Conic section]]
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| | Center of the conic
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| | '''Sturm roulette'''<ref name="sturm">[http://www.mathcurve.com/courbes2d/sturm/sturm.shtml "Sturm's roulette" on www.mathcurve.com]</ref>
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Conic section]]
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| | [[Focus (geometry)|Focus]] of the conic
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| | '''Delaunay roulette'''<ref>[http://www.mathcurve.com/courbes2d/delaunay/delaunay.shtml "Delaunay's roulette" on www.mathcurve.com]</ref>
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Parabola]]
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| | [[Focus (geometry)|Focus]] of the parabola
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| | [[Catenary]]<ref name="2dcurves_roulettede">[http://www.2dcurves.com/roulette/roulettede.html "Delaunay's roulette" on www.2dcurves.com]</ref>
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Ellipse]]
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| | [[Focus (geometry)|Focus]] of the ellipse
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| | '''Elliptic catenary'''<ref name="2dcurves_roulettede"/>
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Hyperbola]]
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| | [[Focus (geometry)|Focus]] of the hyperbola
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| | '''Hyperbolic catenary'''<ref name="2dcurves_roulettede"/>
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Hyperbola]]
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| | [[Centre (geometry)|Center]] of the hyperbola
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| | '''Rectangular elastica'''<ref name="sturm"/>{{Failed verification|date=August 2008}}
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| |-
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| | [[Line (mathematics)|Line]]
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| | [[Epicycloid]] or [[Hypocycloid]]
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| | Center
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| | [[Ellipse]]<ref>[http://www.mathcurve.com/courbes2d/roulette/roulette.shtml "Roulette with straight fixed curve" on www.mathcurve.com]</ref>
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| |-
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| | [[Circle]]
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| | [[Circle]]
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| | ''Any''
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| | [[Centered trochoid]]<ref>[http://www.mathcurve.com/courbes2d/trochoid/trochoidacentre.shtml "Centered trochoid" on mathcurve.com]</ref>
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| |-
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| | [[Parabola]]
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| | Equal parabola parameterized in opposite direction
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| | [[Vertex (curve)|Vertex]] of the parabola
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| | [[Cissoid of Diocles]]<ref name="2dcurves_cubicc">[http://www.2dcurves.com/cubic/cubicc.html "Cissoid" on www.2dcurves.com]</ref>
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| |-
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| | [[Catenary]]
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| | [[Line (mathematics)|Line]]
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| | ''See [[#Example|example]] above''
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| | Line
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| |}
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [[W. H. Besant]] (1890) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=00020002 Notes on Roulettes and Glissettes] from [[Cornell University]] Historical Math Monographs, originally published by Deighton, Bell & Co.
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| *{{MathWorld|urlname=Roulette|title=Roulette}}
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| ==Further reading==
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| *[http://www.2dcurves.com/roulette/roulette.html Roulette at 2dcurves.com]
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| *[http://www.mathcurve.com/courbes2d/base/base.shtml Base, roulante et roulettes d'un mouvement plan sur plan] {{fr icon}}
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| *[http://www.tfh-berlin.de/~schwenk/Lehrgebiete/AUST/Welcome.html Eine einheitliche Darstellung von ebenen, verallgemeinerten Rollbewegungen und deren Anwendungen] {{de icon}}
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| {{Differential transforms of plane curves}}
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| [[Category:Curves]]
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