Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the set of lines intersecting both $B_1$ and $B_2$.
Is there an exact formula for the étendue measure of $C$? To be clear, the “étendue” of a Borel set of lines $C$ in $\mathbb{R}^m$ is defined as one half the integral over unit vectors $u$ in $\mathbb{R}^m$ of the (“area”) measure $\mathcal{A}(C,u)$ of the intersection of an arbitrary hyperplane perpendicular to $u$ with the union of all lines in $C$ having direction $u$. (The factor ½ is to avoid doubly counting each direction.) In other words, this is, up to normalization, the translation-and-rotation invariant measure on the Grassmannian of affine lines in $\mathbb{R}^m$.
For $r_1,r_2$ small with respect to $d$ we should get the asymptotic $V_{n-1}^2 \, r_1^{n-1} \, r_2^{n-1} / d^{n-1}$, where $V_{n-1}$ refers to the volume of the unit $(n-1)$-ball, because we are seeing a small disk of area $V_{n-1} \, r_1^{n-1}$ so solid angle $V_{n-1} \, r_1^{n-1} / d^{n-1}$ from one of are $V_{n-1} \, r_2^{n-1}$ (or vice versa, the situation is symmetric). What I wonder is whether an exact formula can be found.