View source for Folium of Descartes

{{short description|1=Algebraic curve of the form x³ + y³ – 3axy = 0}}
[[File:Kartesisches-Blatt.svg|thumb|250px|right|The folium of Descartes (green) with asymptote (blue) when <math>a = 1</math>]]
In [[geometry]], the '''folium of Descartes''' ({{ety|la|folium|[[leaf]]}}; named for [[René Descartes]]) is an [[algebraic curve]] defined by the [[implicit equation]] 
<math display="block">x^3 + y^3 - 3 a x y = 0.</math>
== History ==
The curve was first proposed and studied by [[René Descartes]] in 1638.<ref name=":0">{{Cite web|last=|first=|date=June 5, 2020|title=Folium of Descartes|url=https://encyclopediaofmath.org/wiki/Folium_of_Descartes|archive-url=|archive-date=|access-date=January 30, 2021|website=Encyclopedia of Mathematics}}</ref> Its claim to fame lies in an incident in the development of [[calculus]]. Descartes challenged [[Pierre de 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]].<ref name=":1">{{Cite book|last=Stewart|first=James|title=Calculus: Early Transcendentals|publisher=Cengage Learning|year=2012|isbn=978-0-538-49790-9|location=United States of America|pages=209–11|chapter=Section 3.5: Implicit Differentiation|author-link=James Stewart (mathematician)}}</ref>

== Graphing the curve ==
[[File:Декартов лист в полярных координатах.png|left|thumb|304x304px|Folium of Descartes in polar coordinates]]
The folium of Descartes can be expressed in [[polar coordinates]] as 
<math display="block">r = \frac{3 a \sin \theta \cos \theta}{\sin^3 \theta + \cos^3 \theta },</math>
which is plotted on the left. This is equivalent to<ref>{{Cite book|last=Stewart|first=James|title=Calculus: Early Transcendentals|publisher=Cengage Learning|year=2012|isbn=978-0-538-49790-9|edition=7th|location=|pages=687|chapter=Chapter 10: Parametric Equations and Polar Coordinates}}</ref>

<math>r = \frac{3a \sec\theta \tan\theta}{1 + \tan^3 \theta}.</math>

Another technique is to write <math>y = px</math> and solve for <math>x</math> and <math>y</math> in terms of <math>p</math>. This yields the [[Rational function|rational]] [[parametric equations]]:<ref>{{cite web | url=https://www.youtube.com/watch?v=jG4DYZ5uuE0  |archive-url=https://ghostarchive.org/varchive/youtube/20211221/jG4DYZ5uuE0 |archive-date=2021-12-21 |url-status=live| title=DiffGeom3: Parametrized curves and algebraic curves | publisher=N J Wildberger, [[University of New South Wales]] | accessdate=5 September 2013}}{{cbignore}}</ref>

: <math>x = {{3ap} \over {1 + p^3}},\, y = {{3ap^2} \over {1 + p^3}}. </math>

We can see that the parameter is related to the position on the curve as follows:
* <math>p < -1</math> corresponds to <math>x>0</math>, <math>y<0</math>: the right, lower, "wing".
* <math>-1 < p < 0</math> corresponds to <math>x<0</math>, <math>y>0</math>: the left, upper "wing".
* <math>p>0</math> corresponds to <math>x>0</math>, <math>y>0</math>: the loop of the curve.

Another way of plotting the function can be derived from symmetry over <math>y = x</math>. 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.

This operation is equivalent to a substitution:
<math display="block"> x = {{u+v} \over {\sqrt{2}}},\, y = {{u-v} \over {\sqrt{2}}} </math>
and yields
<math display="block"> v = \pm u\sqrt{\frac{3a \sqrt{2} - 2u}{6u + 3a \sqrt{2}}}. </math>
Plotting in the Cartesian system of <math>(u,v)</math> gives the folium rotated by 45° and therefore symmetric by <math>u</math>-axis.

== Properties ==
It forms a loop in the first quadrant with a [[double point]] at the origin and [[asymptote]]
<math display="block">x + y + a = 0 \,.</math>
It is symmetrical about the line <math>y = x</math>. As such, the two intersect at the origin and at the point <math>(3a/2,3a/2)</math>.

Implicit differentiation gives the formula for the slope of the tangent line to this curve to be<ref name=":1" /> <math display = block>\frac{dy}{dx} = \frac{ay - x^2}{y^2 - ax}. </math> Using either one of the polar representations above, the area of the interior of the loop is found to be <math> 3a^2/2</math>. Moreover, the area between the "wings" of the curve and its slanted asymptote is also <math>3a^2 / 2</math>.<ref name=":0" />

== Relationship to the trisectrix of Maclaurin ==
[[File:MaclaurinTrisectrix.SVG|thumb|Trisectrix of Maclaurin]]
The folium of Descartes is related to the [[trisectrix of Maclaurin]] by [[affine transformation]]. To see this, start with the equation 
<math display="block">x^3 + y^3 = 3 a x y \,,</math>
and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting  

<math display="block">x = {{X+Y} \over \sqrt{2}}, y = {{X-Y} \over \sqrt{2}}.</math>
In the <math>X,Y</math> plane the equation is 
<math display="block">2X(X^2 + 3Y^2) = 3 \sqrt{2}a(X^2-Y^2).</math>

If we stretch the curve in the <math>Y</math> direction by a factor of <math>\sqrt{3}</math> this becomes 
<math display="block">2X(X^2 + Y^2) = a \sqrt{2}(3X^2-Y^2),</math>
which is the equation of the trisectrix of Maclaurin.

== Notes ==
{{reflist}}

== References ==
* J. Dennis Lawrence: ''A catalog of special plane curves'', 1972, Dover Publications. {{ISBN|0-486-60288-5}}, pp.&nbsp;106–108
* [[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]])

== External links ==
{{commonscat|Folium of Descartes}}
*{{MathWorld|title=Folium of Descartes|urlname=FoliumofDescartes}}
*[http://www-history.mcs.st-andrews.ac.uk/history/Curves/Foliumd.html "Folium of Descartes" at MacTutor's Famous Curves Index]
*[http://www.mathcurve.com/courbes2d.gb/foliumdedescartes/foliumdedescartes.shtml "Cartesian Folium" at MathCurve]

{{Spirals}}

[[Category:René Descartes]]
[[Category:Cubic curves]]