## Cassinian Ovals

Cassinian ovals, also know as Cassini ovals and Cassini ellipses, are a quartic plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted to an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.
The curve was named after Giovanni Cassini, who first investigated it in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun travelled round the Earth on one of these ovals, with the Earth at one focus of the oval. Cassini actually introduced his curves 14 years before Jacob Bernoulli described his lemniscate. The Cassinian ovals are the locus of a point $P$ that moves so that the product of its distances from two fixed points $S$ and T [in this case the points ($\pm$a, 0)] is a constant $b^2$. The shape of the curve depends on $\frac{b}{a}$. If $b>a$ then the curve consists of two loops. If $b<a$ then the curve consists of a single loop. If $a<b$, the area would be $\displaystyle a^2+b^2E\left ( \frac{a^2}{b^2} \right )$, where $E\left ( x \right )$; is the complete elliptic integral of the second kind. If $a=b$, the curve becomes a lemniscate which has an area of $A=2a^2$. If $a>b$, the curve becomes two disjoint ovals which you would have to integrate. Interactive Graph: |
Equations:Cartesian Equation:
$\displaystyle \left ( x^2+y^2 \right )^2-2a^2\left ( x^2 -y^2\right )+a^4-c^b=0$
OR $\displaystyle \left [ \left ( x-a \right )^2+y^2 \right ]\left [ \left ( x+a \right )^2 +y^2\right ]=b^4$ Bipolar Equation:
$\displaystyle r_1 r_2=b^2$
Polar Equation:
$\displaystyle r^4+a^4-2a^2r^2\left [ 1+\cos\left ( 2\theta \right ) \right ]=b^4$
Graphs: |

**References:**

"Cassini Oval."

"Cassinian Ovals."

Weisstein, Eric W. "Cassini Ovals."

*Wikipedia*. Wikimedia Foundation, 15 Nov. 2013. Web. 23 Nov. 2013.*Cassinian Oval*. N.p.: Virtual Mathematics Museum, 17 Nov. 2004. PDF."Cassinian Ovals."

*Cassinian*. N.p., Jan. 1997. Web. 23 Nov. 2013.Weisstein, Eric W. "Cassini Ovals."

*MathWorld*. Wolfram, 25 May 2004. Web. 23 Nov. 2013.