The setting is as follows: we considers 2 disks embedded in $\mathbb R^3,$ and are interested in projecting one disk (either) onto the plane of the other other, and then compute their area of intersection within that chosen plane.

Suppose the plane of the disk ("2") into which we project is $\Pi$ described by a normal vector $\mathbf n,$ and $O_\Pi$ a projection operator, projecting onto an orthonormal basis within the 2D plane (spanned by the null-space of $\mathbf n$). Then if our disk "1" (to be projected onto $\Pi$) is centred at $\mathbf r_1,$ initially in a plane with normal $\mathbf n_1,$ then the half-axis vectors of the projected disk which is an ellipsoid in general ($l$ to denote large axis and $s$ the small one) can be expressed as:

$$ \begin{align} \mathbf a_l =& r (\mathbf n_1 \times \mathbf n) \text{ and projected to } O_\Pi \mathbf a_l \in\mathbb R^2 \tag{1} \\ \mathbf a_s =& r \mathbf n_1 \times (\mathbf n_1 \times \mathbf n) \text{ and similarly } O_\Pi \mathbf a_s\in\mathbb R^2 \tag{2} \end{align} $$

where $r$ is the radius of the two disks. For simplicity, in the plane $\Pi,$ we assume a coordinate system where the original disk of the plane has its centre at origin and the projected disk (the ellipsoid) is centred at $O_\Pi(\mathbf r_1-\mathbf r_2),$ which is the difference vector of the disks' position vectors. To describe the obtained ellipsoid (projected disk), we can use the Kronecker product to compute its covariance matrix $\text{cov}=(O_\Pi a_l \otimes O_\Pi a_l)+(O_\Pi a_s \otimes O_\Pi a_s)$ (whose eigenvalues give the length of the axes and the eigenvectors the orientation), which together with the centre position provide a complete characterisation of the ellipsoid.

For an intuitive picture, here (scrolling down a bit) an example of a disk projected onto another plane, thus the resulting ellipsoid, are visualised. But unlike our problem, there's no other disk lying in the projected plane to consider an intersection with.

- Question is: generally, is there a way to calculate algebraically the intersection area of the disk with the projection (the ellipsoid)? In other words, is the problem of ellipsoid-ellipsoid intersection area in $\mathbb R^2$ solved or is it recommended to recur to numerical schemes? I am not aware of whether there exists algorithms that solve this efficiently.