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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 | |
| :<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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