View source for Descartes oval

A plane curve for which the distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313402.png" /> between any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313403.png" /> of the curve and two fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313405.png" /> (the foci) are related by the non-homogeneous linear equation

<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/d/d031/d031340/d0313406.png" /></td> </tr></table>

A Descartes oval may be defined by means of the homogeneous linear equation

<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/d/d031/d031340/d0313407.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313408.png" /> is the distance to the third focus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313409.png" /> located on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134010.png" />. In the general case, a Descartes oval consists of two closed curves, one enclosing the other (see Fig.). The equation of a Descartes oval in Cartesian coordinates is

<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/d/d031/d031340/d03134011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134012.png" /> is the length of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134015.png" />, the Descartes oval is an ellipse; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134017.png" />, it is a hyperbola; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134018.png" />, it is a [[Pascal limaçon|Pascal limaçon]]. First studied by R. Descartes in the context of problems of optics [[#References|[1]]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031340a.gif" />

Figure: d031340a

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Descartes,   "Géométrie" , Leyden  (1637)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



====Comments====
A Descartes oval is also called a Cartesian oval, or simply a Cartesian.

The [[Caustic|caustic]] of a circle with respect to a point light source in the plane of the circle is the [[Evolute|evolute]] of a Descartes oval.

There are several ways in which a Cartesian oval can arise. One is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134020.png" /> be two cones with axes parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134021.png" />-axis and with circular intersections with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134022.png" />-plane. Then the vertical projection onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134023.png" />-plane of their intersection is a Cartesian oval. This is a result of A. Quetelet. Suppose the apexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134025.png" /> are at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134027.png" />, respectively, and the intersections with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134028.png" />-plane are circles of radius 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134029.png" /> (see below). Then in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134031.png" /> the projection of the intersection is a [[Cardioid|cardioid]].

In [[Polar coordinates|polar coordinates]] the equation for a Cartesian oval is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134032.png" />. Hence when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134033.png" /> the equation becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134034.png" /> (plus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134035.png" />) and one obtains the [[Pascal limaçon|Pascal limaçon]]; when in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134036.png" />, the [[Cardioid|cardioid]] results, whose equation can also be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134037.png" />.

====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>