Q:

Difficult equations to rewrite as ellipses

I have this equality that defines an elliptic boundary.  I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that.  How would I go about rewriting this so it looks like the equation for an ellipse?
$$a + \frac{\sqrt{(p_x - x_x)^2 + (p_y - x_y)^2}}{s_a} = b + \frac{\sqrt{(q_x - x_x)^2 + (q_x - x_y)^2}}{s_b}$$
In this equation, $a, b, s_a, s_b$ are some constants, and $(p_x, p_y), (q_x, q_y), (x_x, x_y)$ are points, where the p and q points are fixed.

A:

In fact, the equation you have is what's generally known as a Cartesian oval.
The equation you gave is similar in form to
$$m \sqrt{(x-a)^2+y^2}\pm n \sqrt{(x+a)^2+y^2}=k$$
which is equation 2 in the MathWorld link I gave, after setting the fixed points $p$ and $q$ (the foci) on the horizontal axis and having the segment joining the two foci be centered at the origin, and identifying $a-b$ or $b-a$ with $k$, and $1/s_a$ and $1/s_b$ with $m$ and $n$.
As noted in the MathWorld link, central conics are degenerate Cartesian ovals, obtained by setting $m=n$.

