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| In [[geometry]], a '''limaçon''' or '''limacon''' {{IPAc-en|ˈ|l|ɪ|m|ə|s|ɒ|n}}, also known as a '''limaçon of Pascal''', is defined as a [[roulette (curve)|roulette]] formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called [[centered trochoid]]s; more specifically, they are [[epitrochoid]]s. The '''[[cardioid]]''' is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a [[cusp (singularity)|cusp]].
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| The term derives from the [[French language|French]] word ''limaçon'', which refers to small [[snail]]s ([[Latin language|Latin]] ''limax''). Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be [[heart]]-shaped, or it may be oval.
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| A limaçon is a [[circular algebraic curve|bicircular]] [[algebraic curve|rational plane algebraic curve]] of degree 4.
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| [[Image:Limacons.svg|thumb|500px|none|Three limaçons: dimpled, with cusp (a [[cardioid]]), and looped, respectively. Not shown: the convex limaçon]]
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| ==History==
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| The earliest formal research on limaçons is generally attributed to [[Étienne Pascal]], father of [[Blaise Pascal]]. However, some insightful investigations regarding them had been undertaken earlier by the [[Germany|German]] [[Renaissance]] artist [[Albrecht Dürer]]. Dürer's ''Underweysung der Messung (Instruction in Measurement)'' contains specific geometric methods for producing limaçons. The curve was named by [[Gilles de Roberval]] when he used it as an example for finding tangent lines.
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| ==Equations==
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| The equation (up to translation and rotation) of a limaçon in [[polar coordinates]] has the form | |
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| :<math>r = b + a \cos \theta \ .</math>
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| This can be converted to [[Cartesian coordinate]]s by multiplying by ''r'' (thus introducing a point at the origin which in some cases is spurious), and substituting <math>r^2 = x^2+y^2</math> and <math>r \, \cos \theta = x</math> to obtain<ref>{{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=113–118 }}</ref>
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| :<math>(x^2+y^2-ax)^2=b^2(x^2+y^2). \,</math>
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| Parametrically, this becomes
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| :<math>x = {a\over 2} + b \cos \theta + {a\over 2} \cos 2\theta,\, y = b \sin \theta + {a\over 2} \sin 2\theta.</math>
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| In the [[complex plane]] this takes the form
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| :<math>z = {a\over 2} + b e^{i\theta} + {a\over 2} e^{2i\theta}.</math>
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| [[Image:EpitrochoidIn1.gif|right|thumb|500px|Construction of a limaçon]]
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| If we shift this horizontally by a/2 we obtain the equation in the usual form for a centered trochoid:
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| :<math>z = b e^{it} + {a\over 2} e^{2it}.</math>
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| This is the equation obtained when the center of the curve (as a centered trochoid) is taken to be the origin.
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| ===Special cases===
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| In the special case a = b, the polar equation is
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| <math> r = b(1 + \cos \theta) = 2b\cos^2 {\theta \over 2}</math> or <math>r^{1 \over 2} = (2b)^{1 \over 2} \cos {\theta \over 2}</math> making it a member of the [[sinusoidal spiral]] family of curves. This curve is the [[cardioid]].
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| In the special case <math>a = 2b</math> the centered trochoid form of the equation becomes
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| :<math>z = b (e^{it} + e^{2it}) = b e^{3it\over 2} (e^{it\over 2} + e^{-it\over 2}) = 2b \cos {t\over 2} e^{3it\over 2} </math>,
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| or, in polar coordinates,
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| :<math>r = 2b\cos{\theta \over 3}</math>
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| making it a member of the [[Rose (mathematics)|rose]] family of curves. This curve is a [[trisectrix]], and is sometimes called the [[limaçon trisectrix]].
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| ==Form==
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| When <math>b > a</math> the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above so the graph of this equation has an [[acnode]] or isolated point.
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| When <math>b > 2a</math> the area bounded by the curve is convex and when <math>a < b < 2a</math> the curve has an indentation bounded by two [[inflection point]]s. At <math>b = 2a</math> the point <math>(-a, 0)</math> is a point of 0 [[curvature]].
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| As <math>b</math> is decreased relative to <math>a</math>, the indentation becomes more pronounced until, at <math>b = a</math>, the cardioid, it becomes a cusp. For <math>0 < b < a</math>, the cusp expands to an inner loop and the curve crosses itself at the origin. As <math>b</math> approaches 0 the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.
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| ==Measurement==
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| The area enclosed by the limaçon <math>r = b + a \cos \theta</math> is <math>(b^2 + {{a^2}\over 2})\pi</math>. When <math>b < a</math> this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles <math>\pi \pm \arccos {b \over a}</math>, the area enclosed by the inner loop is
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| <math>(b^2 + {{a^2}\over 2})\arccos {b \over a} - {3\over 2} b \sqrt {{a^2} - {b^2}}</math>, the area enclosed by the outer loop is <math>(b^2 + {{a^2}\over 2})(\pi - \arccos {b \over a}) + {3\over 2} b \sqrt {{a^2} - {b^2}}</math>, and the area between the loops is <math>(b^2 + {{a^2}\over 2})(\pi - 2\arccos {b \over a}) + 3 b \sqrt {{a^2} - {b^2}}.</math>
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| ==Relation to other curves==
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| * Let P be a point and C be a circle whose center is not P. Then the envelope of those circles whose center lies on C and that pass through P is a limaçon.
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| [[Image:PedalCurve2.gif|500px|right|thumb|Limaçon — pedal curve of a [[circle]]]] | |
| * A [[Pedal curve|pedal]] of a [[circle]] is a limaçon. In fact, the pedal with respect to the origin of the circle with radius <math>b</math> and center <math>(a,0)</math> has polar equation <math>r = b + a \cos \theta</math>
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| * The [[Inversive geometry#Inversion of an algebraic curve|inverse]] with respect to the unit circle of <math>r = b + a \cos \theta</math> is <math>r = {1 \over {b + a \cos \theta}}</math> which is the equation of a conic section with eccentricity a/b and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an ellipse then the corresponding limaçon will have no loop.
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| * The [[conchoid (mathematics)|conchoid]] of a circle with respect to a point on the circle is a limaçon.
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| * A particular special case of a [[Cartesian oval]] is a limaçon.<ref>{{MacTutor|class= Curves|id= Cartesian|title=Cartesian Oval}}</ref>
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| ==References==
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| <references/>
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| ==Additional reading==
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| * Jane Grossman and Michael Grossman. [http://docs.google.com/viewer?a=v&q=cache:jKieaApvrVYJ:poncelet.math.nthu.edu.tw/disk5/js/cardioid/13.pdf+%22dimple+or+no+dimple%22&hl=en&gl=us&pid=bl&srcid=ADGEESiMUIZzRFDZ2Vg6JdJoNh7mABaPjNTwSeJlPV_XYaJeaIjiyFDxO8RWPjEG2j8slKKyRqWDXPWgRZ4RCY1aZfAY8qkkNr1Fxyzy1XsWVDuii1lbAPmQzpl0LHOddHy9ECg_GJ3y&sig=AHIEtbRaIkziXs2lmaxTw7r2zC6LDYiJLw "Dimple or no dimple"], ''The Two-Year College Mathematics Journal'', January 1982, pages 52–55.
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| * Howard Anton. ''Calculus'', 2nd edition, page 708, John Wiley & Sons, 1984.
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| * Howard Anton. [http://higheredbcs.wiley.com/legacy/college/anton/0471472441/add_material/analytic_geometry_in_calculus.pdf] pp. 725 – 726.
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| * Howard Eves. ''A Survey of Geometry'', Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.
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| ==External links==
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| * [http://mathworld.wolfram.com/Limacon.html Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web Resource.]
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| * [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Limacon.html "Limacon of Pascal" at The MacTutor History of Mathematics archive]
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| * [http://www.2dcurves.com/roulette/roulettel.html "Limaçon" at www.2dcurves.com]
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| * [http://www.mathcurve.com/courbes2d/limacon/limacon.shtml "Limaçons de Pascal" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
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| * [http://xahlee.org/SpecialPlaneCurves_dir/LimaconOfPascal_dir/limaconOfPascal.html "Limacon of Pascal" at Visual Dictionary of Special Plane Curves]
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| * [http://communities.ptc.com/videos/2080 "Limacon of Pascal" on PlanetPTC (Mathcad)]
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| ==See also==
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| * [[List of periodic functions]]
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| {{DEFAULTSORT:Limacon}}
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| [[Category:Algebraic curves]]
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