Two-dimensional visibility charts for continuous curves

This paper considers computation of visibility for two-dimensional shapes whose boundaries are C1 continuous curves. We assume we are given a one-parameter family of candidate viewpoints, which may be interior or exterior to the object, and at finite or infinite locations. We consider how to compute whether the whole boundary of the shape is visible from some finite set of viewpoints taken from this family, and if so, how to compute a minimal set of such viewpoints. The viewpoint families we handle include (i) the set of viewing directions from infinity, (ii) viewpoints on a circle located outside the object (for inspection from a turntable), and (iii) viewpoints located on the walls of the shape itself. We compute a structure called a visibility chart, which simultaneously encodes the visible part of the shape's boundary from every view in the family. Using such a visibility chart, finding a minimal set of viewpoints reduces to the set-covering problem over the reals. Practical algorithms are obtained by a discrete sampling of the visibility chart. For exterior visibility problems, a reasonable approach is to compute an almost-optimal solution (in terms of number of viewpoints), which can be done in almost-linear time. For interior visibility problems, or when a more correct solution is required, we solve the general set-covering problem, guaranteeing an optimal solution but taking exponential time.