  The essential variety is an algebraic subvariety of dimension $5$ in real
projective space $\mathbb R\mathrm P^{8}$ which encodes the relative pose of
two calibrated pinhole cameras. The $5$-point algorithm in computer vision
computes the real points in the intersection of the essential variety with a
linear space of codimension $5$. The degree of the essential variety is $10$,
so this intersection consists of 10 complex points in general.
  We compute the expected number of real intersection points when the linear
space is random. We focus on two probability distributions for linear spaces.
The first distribution is invariant under the action of the orthogonal group
$\mathrm{O}(9)$ acting on linear spaces in $\mathbb R\mathrm P^{8}$. In this
case, the expected number of real intersection points is equal to $4$. The
second distribution is motivated from computer vision and is defined by
choosing 5 point correspondences in the image planes $\mathbb R\mathrm
P^2\times \mathbb R\mathrm P^2$ uniformly at random. A Monte Carlo computation
suggests that with high probability the expected value lies in the interval
$(3.95 - 0.05,\ 3.95 + 0.05)$.
