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| [[File:FoliumDescartes.svg|thumb|250px|right|The folium of Descartes (black) with asymptote (blue).]]
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| In [[geometry]], the '''folium of Descartes''' is an [[algebraic curve]] defined by the equation
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| :<math>x^3 + y^3 - 3 a x y = 0 \,</math>.
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| It forms a loop in the first quadrant with a [[double point]] at the origin and [[asymptote]]
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| :<math>x + y + a = 0 \,</math>.
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| It is symmetrical about <math>y = x</math>.
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| The name comes from the [[Latin]] word ''folium'' which means "[[leaf]]".
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| The curve was featured, along with a portrait of Descartes, on an Albanian stamp in 1966.
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| == History ==
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| The curve was first proposed by [[Descartes]] in 1638. Its claim to fame lies in an incident in the development of [[calculus]]. Descartes challenged [[Fermat]] to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.<ref>Simmons, p. 101</ref> Since the invention of calculus, the slope of the tangent line can be found easily using [[implicit differentiation]].
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| == Graphing the curve ==
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| Since the equation is degree 3 in both x and y, and does not factor, it is difficult to solve for one of the variables.
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| However, the equation in [[polar coordinates]] is:
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| :<math>r = \frac{3 a \sin \theta \cos \theta}{\sin^3 \theta + \cos^3 \theta }.</math>
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| which can be plotted easily.
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| Another technique is to write y = px and solve for x and y in terms of p. This yields the [[Rational function|rational]] [[parametric equations]]:<ref>{{cite web | url=http://www.youtube.com/watch?v=jG4DYZ5uuE0 | title=DiffGeom3: Parametrized curves and algebraic curves | publisher=N J Wildberger, [[University of New South Wales]] | accessdate=5 September 2013}}</ref>
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| <math>x = {{3ap} \over {1 + p^3}},\, y = {{3ap^2} \over {1 + p^3}}</math>.
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| We can see that the parameter is related to the position on the curve as follows:
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| * ''p'' < -1 corresponds to x>0, y<0: the right, lower, "wing".
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| * -1 < ''p'' < 0 corresponds to x<0, y>0: the left, upper "wing".
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| * ''p'' > 0 corresponds to x>0, y>0: the loop of the curve.
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| Another way of plotting the function can be derived from symmetry over y = x. The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis.
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| This operation is equivalent to a substitution:
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| :<math> x = {{u+v} \over {\sqrt{2}}},\, y = {{u-v} \over {\sqrt{2}}} </math>
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| and yields
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| :<math> v = \pm u\sqrt{\frac{3\sqrt{2} - 2u}{6u + 3\sqrt{2}}} </math>
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| Plotting in the cartesian system of (u,v) gives the folium rotated by 45° and therefore symmetric by u axis.
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| == Relationship to the trisectrix of MacLaurin ==
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| The folium of Descartes is related to the [[trisectrix of Maclaurin]] by [[affine transformation]]. To see this, start with the equation
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| :<math>x^3 + y^3 = 3 a x y \,</math>,
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| and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting <math>x = {{X+Y} \over \sqrt{2}}, y = {{X-Y} \over \sqrt{2}}</math>. In the <math>X,Y</math> plane the equation is
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| :<math>2X(X^2 + 3Y^2) = 3 \sqrt{2}a(X^2-Y^2)</math>.
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| If we stretch the curve in the <math>Y</math> direction by a factor of <math>\sqrt{3}</math> this becomes
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| :<math>2X(X^2 + Y^2) = a \sqrt{2}(3X^2-Y^2)</math>
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| which is the equation of the trisectrix of Maclaurin.
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * J. Dennis Lawrence: ''A catalog of special plane curves'', 1972, Dover Publications. ISBN 0-486-60288-5, pp. 106–108
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| * George F. Simmons: ''Calculus Gems: Brief Lives and Memorable Mathematics'', New York 1992, McGraw-Hill, xiv,355. ISBN 0-07-057566-5; new edition 2007, The Mathematical Association of America ([[The Mathematical Association of America|MAA]])
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| == External links ==
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| {{commonscat|Folium of Descartes}}
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| * [http://www.mindspring.com/~r.amoroso/Amoroso7.pdf Richard L. Amoroso: ''Fe, Fi, Fo, Folium: A Discourse on Descartes’ Mathematical Curiosity'']
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| *{{MathWorld|title=Folium of Descartes|urlname=FoliumofDescartes}}
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| *[http://www-history.mcs.st-andrews.ac.uk/history/Curves/Foliumd.html "Folium of Descartes" at MacTutor's Famous Curves Index]
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| *[http://www.mathcurve.com/courbes2d/foliumdedescartes/foliumdedescartes.shtml "Folium de Descartes" at Encyclopédie des Formes Mathématiques Remarquables]
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| [[Category:Curves]]
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| [[Category:Algebraic curves]]
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| [[Category:René Descartes]]
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Myrtle Benny is how I'm known as and I feel comfy when people use the complete name. What I adore doing is playing baseball but I haven't made a dime with it. California is where her home is but she requirements to move because of her family members. Bookkeeping is her day job now.
Feel free to visit my weblog; at home std test (over at this website)