Average degree of the essential variety!

The essential variety is an algebraic subvariety of dimension $5$ in $\mathbb{R}{P}^3.$ It encodes the relative pose of two calibrated cameras, where a calibrated camera is a matrix of the form $[R,t]$ with $R\in SO(3)$ and $t\in {\mathbb{R}}^3$. Since the degree of this variety is $10$, there can only be at most $10$ complex solutions. We compute the expected number of real points in the intersection of the essential variety with a random linear space of codimension $5$. My aim is to tell you about these computations and our results. This is joint work with Paul Breiding, Samantha Fairchild, and Pierpaola Santarsiero.