View source for Folium of Descartes

A plane algebraic curve of order three which is given in Cartesian coordinates by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407501.png" />; the parametric equations are

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407502.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407503.png" /> is the tangent of the angle between the radius vector of the curve and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407504.png" />-axis. The folium of Descartes is symmetric about the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407505.png" /> (see Fig.). The tangent lines are parallel to the coordinate axes at the points with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407507.png" />. The coordinate origin is a nodal point with the coordinate axes as tangent lines. The asymptote is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407508.png" />. The surface area enclosed between the curve and the asymptote is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f0407509.png" />. The surface area of the loop is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040750/f04075010.png" />. Named after R. Descartes who was the first to study it in 1638.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040750a.gif" />

Figure: f040750a

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,   E.S. Stolova,   "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,   "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR></table>