Ray model in $3D$

Question

Supposing we have a point source $p$ and multiple receiving points $r_i$ in $\Bbb R^3$ and there is a direct ray from $p$ to each $r_i$ and if there is a single mirror there is a single reflected ray from $p$ to each $r_i$. Is there an elementary proof the reflected rays when extrapolated behind the mirror should converge? In $2D$ this is canonical since there is one plane to handle. In $3D$ each $(p,r_i)$ and $(p,r_j)$ lie on a distinct plane and this makes it non-intuitive.

Answer 1

Let $M$ be the plane of the mirror, $\bar{p}$ the image of $p$ under reflection in $M$, and $\bar{r}_{i}$ the image of $r_{i}$ under reflection in $M$.

For each $i$, the points $p$, $\bar{p}$, $r_{i}$, and $\bar{r}_{i}$ lie in a plane orthogonal to $M$ (because the segments $p\bar{p}$ and $r_{i}\bar{r}_{i}$ are orthogonal to $M$), and the reflected ray from $p$ to $M$ to $r_{i}$ corresponds in the obvious way with the line segment from $\bar{p}$ to $r_{i}$, by the two-dimensional case.

Consequently, the rays from the $r_{i}$ that reflect to $p$ do in fact converge at $\bar{p}$ behind the mirror.