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 $(3.95 - 0.05,\ 3.95 + 0.05)$.

翻译：本质簇是实射影空间 $\mathbb R\mathrm P^{8}$ 中一个维数为 $5$ 的代数子簇，它编码了两个已标定针孔相机的相对位姿。计算机视觉中的五点算法计算本质簇与余维数 $5$ 的线性空间交集中的实点。本质簇的次数为 $10$，因此一般情况下该交集包含 $10$ 个复点。我们计算了当线性空间为随机时实交点的期望数量。我们关注线性空间的两种概率分布。第一种分布在正交群 $\mathrm{O}(9)$ 作用于 $\mathbb R\mathrm P^{8}$ 中线性空间的作用下保持不变。在此情形下，实交点的期望数量等于 $4$。第二种分布源于计算机视觉，通过在像平面 $\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$ 中均匀随机选取 5 组点对应来定义。蒙特卡洛计算表明，期望值以高概率落在区间 $(3.95 - 0.05,\ 3.95 + 0.05)$ 内。