Let two ellipsoids are
$$\tag{1} \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} + \frac{(z-z_0)^2}{c^2} =1$$ $$ \tag{2} \frac{x^2}{\ell^2} + \frac{y^2+z^2}{\ell^2-c^2} =1$$
To find the intersection curve of these two ellipsoids, I let $$ x = \ell sin\theta cos\phi $$ $$ y = \sqrt{\ell^2-c^2} sin\theta sin\phi $$ $$ z = \sqrt{\ell^2-c^2} cos\theta $$
Then, substitute the above $x,y,z$ into (1) and try to solve a quartic equation. Thus, I can attain $\theta$ as a function of $\phi$ and plot the intersection curve easily through $\phi = 0 \sim 2\pi$.
However,the quartic formula is too unwieldy, so is there any other idea to solve this problem? or useful approximation and expression that I can use?