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en>Fustis: /* Mathematics */

2014-12-10T11:14:56Z

Mathematics

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	Jerrie is what you ~~could certainly~~ call me ~~but My family and i don't like when~~ people ~~use~~ my ~~full identify~~. ~~The job~~ I~~'ve been occupying for years~~ is ~~a real people manager. Guam is even~~ I'~~ve always been difficult~~. ~~What I truly doing is fish keeping~~ and I~~'ll be [http://data.Gov.uk/data/search?q=creating creating] something else along with it~~. Go to ~~some of~~ my website to find ~~playing~~ more: http://~~circuspartypanama~~.~~com~~<br><br>~~Also visit~~ my ~~homepage ...~~ [http://~~circuspartypanama.com clash of clans hack tool v8~~.8]		Jerrie Swoboda is what you can call me so I totally dig where it name. Managing people is my day of the week job now. As a girl what I do like is to master croquet but I can't make it my position really. My spouse and I chose to call home in Massachusetts. Go to my website to find out more: http://prometeu.net<br><br>Take a look at my blog - [http://prometeu.net Clash Of Clans Hack Tool]

en>Widr: Reverted edits by 81.104.9.109 (talk) to last version by Beland

2014-03-04T16:57:41Z

Reverted edits by 81.104.9.109 (talk) to last version by Beland

← Older revision		Revision as of 18:57, 4 March 2014
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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]].		Jerrie is what you could certainly call me but My family and i don't like when people use my full identify. The job I've been occupying for years is a real people manager. Guam is even I've always been difficult. What I truly doing is fish keeping and I'll be [http://data.Gov.uk/data/search?q=creating creating] something else along with it. Go to some of my website to find playing more: http://circuspartypanama.com<br><br>Also visit my homepage ... [http://circuspartypanama.com clash of clans hack tool v8.8]

	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.~~

	~~A limaçon is a [[circular algebraic curve\|bicircular]] [[algebraic curve\|rational plane algebraic curve]] of degree 4.~~

	~~[[Image:Limacons.svg\|thumb\|500px\|none\|Three limaçons: dimpled, with cusp (a [[cardioid]]), and looped, respectively. Not shown: the convex limaçon]]~~

	~~==History==~~
	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~~.

	~~==Equations==~~
	The ~~equation (up to translation and rotation) of a limaçon in [[polar coordinates]] has the form~~

	~~:<math>r = b + a \cos \theta \ .</math>~~

	~~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>

	~~:<math>(x^2+y^2-ax)^2=b^2(x^2+y^2). \,</math>~~

	~~Parametrically, this becomes~~

	~~:<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>~~

	~~In the [[complex plane]] this takes the form~~

	~~:<math>z = {a\over 2} + b e^{i\theta} + {a\over 2} e^{2i\theta}.</math>~~

	~~[[Image:EpitrochoidIn1.gif\|right\|thumb\|500px\|Construction of a limaçon]]~~

	~~If we shift this horizontally by a/2 we obtain the equation in the usual form~~ for ~~a centered trochoid:~~
	~~:<math>z = b e^{it} + {a\over 2} e^{2it}.</math>~~

	~~This~~ is ~~the equation obtained when the center of the curve (as~~ a ~~centered trochoid) is taken to be the origin~~.

	~~===Special cases===~~
	~~In the special case a = b, the polar equation~~ is
	~~<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]].~~

	~~In the special case <math>a = 2b</math> the centered trochoid form of the equation becomes~~
	~~:<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>,~~

	~~or, in polar coordinates,~~

	~~:<math>r = 2b\cos{\theta \over 3}</math>~~

	~~making it a member of the [[Rose (mathematics)\|rose]] family of curves. This curve is a [[trisectrix]],~~ and ~~is sometimes called the [[limaçon trisectrix]].~~

	~~==Form==~~
	~~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.~~

	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]].

	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.

	~~==Measurement==~~
	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
	<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>

	~~==Relation to other curves==~~
	* 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.~~
	[~~[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>
	* 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.
	* The [[conchoid (mathematics)\|conchoid]] of a circle with respect to a point on the circle is a limaçon.
	* A particular special case of a [[Cartesian oval]] is a limaçon.<ref>{{MacTutor\|class= Curves\|id= Cartesian\|title=Cartesian Oval}}</~~ref>~~

	~~==References==~~
	~~<references~~/>

	~~==Additional reading==~~
	* 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~~.
	* Howard Anton. ''Calculus'', 2nd edition, page 708, John Wiley & Sons, 1984.
	* Howard Anton. [http://higheredbcs.wiley.com/legacy/college/anton/0471472441/add_material/analytic_geometry_in_calculus.pdf] pp. 725 – 726.
	* Howard Eves. ''A Survey of ~~Geometry'', Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.~~

	~~==External links==~~
	* [http://mathworld.wolfram.com/Limacon.html Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web Resource.]
	* [http:~~//www-history.mcs.st-andrews.ac.uk/history/Curves/Limacon.html "Limacon of Pascal" at The MacTutor History of Mathematics archive]~~
	* [http://~~www.2dcurves~~.com~~/roulette/roulettel~~.~~html "Limaçon" at www~~.~~2dcurves~~.~~com]~~
	* [http://~~www.mathcurve~~.com~~/courbes2d/limacon/limacon.shtml "Limaçons de Pascal" at Encyclopédie des Formes Mathématiques Remarquables] (in French)~~
	* [http://xahlee.org/SpecialPlaneCurves_dir/LimaconOfPascal_dir/limaconOfPascal.html "Limacon of Pascal" at Visual Dictionary of ~~Special Plane Curves]~~
	* [http://communities.ptc.~~com/videos/2080 "Limacon of Pascal" on PlanetPTC (Mathcad)]~~

	~~==See also==~~

	* [[List of periodic functions]]

	~~{{DEFAULTSORT:Limacon}}~~
	~~[[Category:Algebraic curves]~~]

en>John of Reading: Typo/general fixing, replaced: Since 1990s → Since the 1990s using AWB

2014-02-03T13:42:42Z

Typo/general fixing, replaced: Since 1990s → Since the 1990s using AWB

← Older revision		Revision as of 15:42, 3 February 2014
Line 1:		Line 1:
	~~Hi there!~~ :) ~~My name~~ is ~~Cecelia~~, I'm a ~~student studying Anthropology~~ and ~~Sociology from Bordeaux~~, ~~France~~.<br><br>~~Stop~~ by ~~my homepage~~ - [http://~~www~~.~~gauhaticommercecollege~~.in/~~content~~/~~turn~~-college-~~success~~-~~story~~-~~these~~-~~promo-codes-bookbyte Bookbyte Textbooks Coupons~~]		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]].

			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.

			A limaçon is a [[circular algebraic curve\|bicircular]] [[algebraic curve\|rational plane algebraic curve]] of degree 4.

			[[Image:Limacons.svg\|thumb\|500px\|none\|Three limaçons: dimpled, with cusp (a [[cardioid]]), and looped, respectively. Not shown: the convex limaçon]]

			==History==
			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.

			==Equations==
			The equation (up to translation and rotation) of a limaçon in [[polar coordinates]] has the form

			:<math>r = b + a \cos \theta \ .</math>

			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>

			:<math>(x^2+y^2-ax)^2=b^2(x^2+y^2). \,</math>

			Parametrically, this becomes

			:<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>

			In the [[complex plane]] this takes the form

			:<math>z = {a\over 2} + b e^{i\theta} + {a\over 2} e^{2i\theta}.</math>

			[[Image:EpitrochoidIn1.gif\|right\|thumb\|500px\|Construction of a limaçon]]

			If we shift this horizontally by a/2 we obtain the equation in the usual form for a centered trochoid:
			:<math>z = b e^{it} + {a\over 2} e^{2it}.</math>

			This is the equation obtained when the center of the curve (as a centered trochoid) is taken to be the origin.

			===Special cases===
			In the special case a = b, the polar equation is
			<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]].

			In the special case <math>a = 2b</math> the centered trochoid form of the equation becomes
			:<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>,

			or, in polar coordinates,

			:<math>r = 2b\cos{\theta \over 3}</math>

			making it a member of the [[Rose (mathematics)\|rose]] family of curves. This curve is a [[trisectrix]], and is sometimes called the [[limaçon trisectrix]].

			==Form==
			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.

			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]].

			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.

			==Measurement==
			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
			<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>

			==Relation to other curves==
			* 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.
			[[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>
			* 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.
			* The [[conchoid (mathematics)\|conchoid]] of a circle with respect to a point on the circle is a limaçon.
			* A particular special case of a [[Cartesian oval]] is a limaçon.<ref>{{MacTutor\|class= Curves\|id= Cartesian\|title=Cartesian Oval}}</ref>

			==References==
			<references/>

			==Additional reading==
			* 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.
			* Howard Anton. ''Calculus'', 2nd edition, page 708, John Wiley & Sons, 1984.
			* Howard Anton. [http://higheredbcs.wiley.com/legacy/college/anton/0471472441/add_material/analytic_geometry_in_calculus.pdf] pp. 725 – 726.
			* Howard Eves. ''A Survey of Geometry'', Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.

			==External links==
			* [http://mathworld.wolfram.com/Limacon.html Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web Resource.]
			* [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Limacon.html "Limacon of Pascal" at The MacTutor History of Mathematics archive]
			* [http://www.2dcurves.com/roulette/roulettel.html "Limaçon" at www.2dcurves.com]
			* [http://www.mathcurve.com/courbes2d/limacon/limacon.shtml "Limaçons de Pascal" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
			* [http://xahlee.org/SpecialPlaneCurves_dir/LimaconOfPascal_dir/limaconOfPascal.html "Limacon of Pascal" at Visual Dictionary of Special Plane Curves]
			* [http://communities.ptc.com/videos/2080 "Limacon of Pascal" on PlanetPTC (Mathcad)]

			==See also==

			* [[List of periodic functions]]

			{{DEFAULTSORT:Limacon}}
			[[Category:Algebraic curves]]

146.179.8.172: mathematic -> mathematical

2012-08-22T13:46:23Z

mathematic -> mathematical

New page

Hi there! :) My name is Cecelia, I'm a student studying Anthropology and Sociology from Bordeaux, France.<br><br>Stop by my homepage - [http://www.gauhaticommercecollege.in/content/turn-college-success-story-these-promo-codes-bookbyte Bookbyte Textbooks Coupons]