Parametric equation of ellipse given by its foci and sum distance

Question

I am having a problem with converting the equation of a general (tilted) ellipse, from its geometric form to a parametric form:

$\sqrt{(x-f_1)^2+(y-f_2)^2} + \sqrt{(x-g_1)^2 + (y-g_2)^2}=S$

where the focal points are $F=(f_1, f_2)$ and $G=(g_1, g_2)$, and the sum distance is $S$.

I would like to have a parametric set of equations for $x(t)$ and $y(t)$ which would also depend only on the focal points $F, G$, and the sum distance $S$.

Answer 1

Thanks to @Dylan for the idea. Here is my solution which seems to work:

$x(t)=a\cos(t)\cos\theta+b\sin(t)\sin\theta+c_1$

$y(t)=-a\cos(t)\sin\theta+b\sin(t)\cos\theta+c_2$,

where, in terms of the focal points $F=(f_1, f_2)$, $G=(g_1, g_2)$, and the sum distance $S$, the parameters are:

$a = \frac{S}{2}$,

$b = \frac{1}{2}\sqrt{S^2-(f_1-g_1)^2-(f_2-g_2)^2}$,

$\theta = \arctan\frac{g_2-f_2}{f_1-g_1}$,

and

$C=(c_1, c_2)=\left(\frac{f_1+g_1}{2}, \frac{f_2+g_2}{2}\right)$.

Of course, there is a condition on $S > \sqrt{(f_1-g_1)^2+(f_2-g_2)^2}$, for the ellipse to exist.