## Cartesian Oval

The "Cartesian ovals," sometimes also known as the Cartesian curve or oval of Descartes, are the quartic curve consisting of two ovals.
It is the locus of a point P whose distances from two foci $F_1$ and $F_2$ in two-center bipolar coordinates satisfy $\displaystyle mr\pm nr'=k$ where $m$, $n$ are positive integers, $k$ is a positive real number, and $r$ and $r'$ are distances from $F_1$ and $F_2$. Cartesian ovals are anallagmatic curves. Unlike the Cartesian ovals, these curves possess three foci. The curves were first studied by Descartes in 1637 and are sometimes called the 'Ovals of Descartes'. The curve was also studied by Newton in his classification of cubic curves. If $m = 1$ then the Cartesian Oval C is a central conic while if $\displaystyle m=\frac{a}{c}$ then the curve is a Limacon of Pascal (Étienne Pascal). In this case the inside oval touches the outside one. Interactive Graphic: |
Equations:Cartesian Equation:
OR
$\displaystyle m\sqrt{\left ( x-a \right )^2+y^2}+2\sqrt{\left ( x+a \right )^2+y^2}=k$ Graph: |

**References:**

"Cartesian Oval."

"Cartesian Oval."

Weisstein, Eric W. "Cartesian Ovals."

*Cartesian*. N.p., Jan. 1997. Web. 16 Nov. 2013."Cartesian Oval."

*Wikipedia*. Wikimedia Foundation, 24 Sept. 2013. Web. 16 Nov. 2013.Weisstein, Eric W. "Cartesian Ovals."

*MathWorld*. Wolfram, 7 July 2010. Web. 16 Nov. 2013.