Suppose I have two ellipsoids in $\Bbb{R^3}$, with one centered at the origin and aligned to the coordinate axis, and the other in general position. I can represent these by:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ $$ \left[\vec{x}-\vec{v}\right]^TA\left[\vec{x}-\vec{v}\right] - 1 = 0 $$
where $\vec{x} = \left[x\ y\ z\ \right]^T$, $\vec{v} = \left[x_0\ y_0\ z_0\ \right]^T$ is the center of the ellipsoid in the general position, and $A = WGW^T$ such that $W$ is a matrix of the unit semi-axes of the ellipsoid as column vectors, and $G$ is a diagonal matrix containing the inverse of the squares of the length of the corresponding semi-axes of the ellipsoid along the diagonal.
The intersection of these two surfaces, if it exists, is a curve. This curve may be continuous, or contain two different segments, depending on whether one is partially "inside" the other.
Assuming that we have a continuous curve, and assuming we know two points on the curve, I would like to find the length of the curve between these two points.
From what I understand, to compute this I first need to find a suitable parametrization for the curve. However, I am struggling with how to go about doing this.
Solving for $z$ in one and then substituting into the other leads to a messy quartic of two variables, and it is not clear how to separate the variables or what parametrization would apply.
I tried expressing one in terms of spherical coordinates $\theta$ and $\phi$, and then substituting into the other. I still get an equation where it is not clear how to separate $\theta$ and $\phi$ and how I would go about parametrizing the result.
Am I approaching this the right way? How would I go about solving this problem?
Thanks!