An Algorithm for Computing Constrained Reflection Paths in Simple Polygon

Let $s$ be a source point and $t$ be a destination point inside an $n$-vertex simple polygon $P$. Euclidean shortest paths and minimum-link paths between $s$ and $t$ inside $P$ have been well studied. Both these kinds of paths are simple and piecewise-convex. However, computing optimal paths in the context of diffuse or specular reflections does not seem to be an easy task. A path from a light source $s$ to $t$ inside $P$ is called a diffuse reflection path if the turning points of the path lie in the interiors of the boundary edges of $P$. A diffuse reflection path is said to be optimal if it has the minimum number of turning points amongst all diffuse reflection paths between $s$ and $t$. The minimum diffuse reflection path may not be simple. The problem of computing the minimum diffuse reflection path in low degree polynomial time has remained open. In our quest for understanding the geometric structure of the minimum diffuse reflection paths vis-a-vis shortest paths and minimum link paths, we define a new kind of diffuse reflection path called a constrained diffuse reflection path where (i) the path is simple, (ii) it intersects only the eaves of the Euclidean shortest path between $s$ and $t$, and (iii) it intersects each eave exactly once. For computing a minimum constrained diffuse reflection path from $s$ to $t$, we present an $O(n(n+\beta))$ time algorithm, where $\beta =\Theta (n^2)$ in the worst case. Here, $\beta$ depends on the shape of the polygon. We also establish some properties relating minimum constrained diffuse reflection paths and minimum diffuse reflection paths. Constrained diffuse reflection paths introduced in this paper provide new geometric insights into the hitherto unknown structures and shapes of optimal reflection paths.