The folium of Descartes (green) with asymptote (blue).
In geometry, the folium of Descartes is an algebraic curve defined by the equation

x^3 + y^3  3 a x y = 0 \,.
It forms a loop in the first quadrant with a double point at the origin and asymptote

x + y + a = 0 \,.
It is symmetrical about y = x.
The name comes from the Latin word folium which means "leaf".
The curve was featured, along with a portrait of Descartes, on an Albanian stamp in 1966.
Contents

History 1

Graphing the curve 2

Relationship to the trisectrix of MacLaurin 3

Notes 4

References 5

External links 6
History
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.^{[1]} Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.
Graphing the curve
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.
However, the equation in polar coordinates is:

r = \frac{3 a \sin \theta \cos \theta}{\sin^3 \theta + \cos^3 \theta }.
which can be plotted easily.
Another technique is to write y = px and solve for x and y in terms of p. This yields the rational parametric equations:^{[2]}
x = ,\, y = .
We can see that the parameter is related to the position on the curve as follows:

p < 1 corresponds to x>0, y<0: the right, lower, "wing".

1 < p < 0 corresponds to x<0, y>0: the left, upper "wing".

p > 0 corresponds to x>0, y>0: the loop of the curve.
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.
This operation is equivalent to a substitution:

x = },\, y = }
and yields

v = \pm u\sqrt{\frac{3a \sqrt{2}  2u}{6u + 3a \sqrt{2}}}
Plotting in the cartesian system of (u,v) gives the folium rotated by 45° and therefore symmetric by u axis.
Relationship to the trisectrix of MacLaurin
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation

x^3 + y^3 = 3 a x y \,,
and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting x = , y = . In the X,Y plane the equation is

2X(X^2 + 3Y^2) = 3 \sqrt{2}a(X^2Y^2).
If we stretch the curve in the Y direction by a factor of \sqrt{3} this becomes

2X(X^2 + Y^2) = a \sqrt{2}(3X^2Y^2)
which is the equation of the trisectrix of Maclaurin.
Notes

^ Simmons, p. 101

^ "DiffGeom3: Parametrized curves and algebraic curves". N J Wildberger,
References

J. Dennis Lawrence: A catalog of special plane curves, 1972, Dover Publications. ISBN 0486602885, pp. 106–108

George F. Simmons: Calculus Gems: Brief Lives and Memorable Mathematics, New York 1992, McGrawHill, xiv,355. ISBN 0070575665; new edition 2007, The Mathematical Association of America (MAA)
External links

Fe, Fi, Fo, Folium: A Discourse on Descartes’ Mathematical CuriosityRichard L. Amoroso:

Weisstein, Eric W., "Folium of Descartes", MathWorld.

"Folium of Descartes" at MacTutor's Famous Curves Index

"Folium de Descartes" at Encyclopédie des Formes Mathématiques Remarquables
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.