I've already worked out the intersection of two circles, now I'm trying to expand / generalize that to the intersection of 2 ellipses (if the foci are coincident, then they're circles), and then to spheres / ellipsoids.
I've been having a devil of a time finding information on this, and unraveling the equations is giving me quite the challenge.
Pointers please? Thanks!
Intersecting two ellipses is much harder than intersecting two circles. There are four possible points of intersection, so you can expect to end up solving a fourth-degree equation.
Here's one way to do it:
Set up a coordinate system that's aligned with the axes of the first ellipse, $E1$. Then, in this coordinate system, you can write simple parametric equations for $E1$. For example: $$ x = \frac{a(1-t^2)}{1+t^2} \quad ; \quad y = \frac{2bt}{1+t^2} $$ In this same coordinate system, write ellipse $E2$ in implicit form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Substitute $x$ and $y$ from $E1$ into this implicit equation, and you'll get a fourth degree equation in $t$. The solutions of this equation are the $t$ values at which intersections occur.
Caveat: the above isn't quite correct, because the parameterisation I gave for ellipse $E1$ doesn't cover the point $(-a,0)$. You can either specifically test this point, to see if it's an intersection, or split the ellipse into two and parameterize each half separately.
Another approach:
Let's call the two ellipses $P(x,y)=0$ and $Q(x,y)=0$. Let $R = P + \lambda Q$. Then the equation $R(x,y) = 0$ also represents a conic section curve of some sort, and, for any value of $\lambda$, this curve passes through the intersection points of $P=0$ and $Q=0$. By a judicious choice of $\lambda$, we can cause the curve $R=0$ to become degenerate (i.e. to become a pair of straight lines). Then all we need to do is intersect these two straight lines with one of the original ellipses. This is fundamentally the same as the approach presented by Levin in the paper referenced below.
Spheres are easy -- the intersections are just circles.
Ellipsoids are difficult, and (depending on your needs), you'll probably be better off using some sort of spline approximation of the intersection curve(s). The first paper I know about was this one by Levin. Then here is a fairly recent paper on the subject.
See also this answer.
