# Folium of Descartes

From Encyclopedia of Mathematics

A plane algebraic curve of order three which is given in Cartesian coordinates by the equation ; the parametric equations are

where is the tangent of the angle between the radius vector of the curve and the -axis. The folium of Descartes is symmetric about the axis (see Fig.). The tangent lines are parallel to the coordinate axes at the points with coordinates and . The coordinate origin is a nodal point with the coordinate axes as tangent lines. The asymptote is given by . The surface area enclosed between the curve and the asymptote is . The surface area of the loop is . Named after R. Descartes who was the first to study it in 1638.

Figure: f040750a

#### References

[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |

[2] | A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian) |

#### Comments

#### References

[a1] | J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) |

[a2] | K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962) |

**How to Cite This Entry:**

Folium of Descartes. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Folium_of_Descartes&oldid=13430

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098