Realizable Configurations of Lines in Pictures of Polyhedrat

In an idealized picture of a scene that contains only polyhedra each line segment that is recorded can have only one of four possible "meanings". In order to understand the picture it is necessary that we be able to label each line with one of the four corresponding labels: +, or A "+" or "." label is associated, respectively, with a convex or concave edge that has both of its two associated planes visible. A line labelled with an arrow refers to a convex edge oriented so that only one of these two planes is visible from the camera and the other is hidden behind it. The orientation of the arrow along the line is such that the planes are to the right of the arrow. If no consistent set of line labels is possible the picture is of an "impossible"--object. If one or more labellings are possible a necessary condition for the picture to be realizable will have been satisfied and the picture may, indeed, be ambiguous. An earlier paper (Huffman, 1971) dealt with the restricted case of scenes containing only trihedral objects. This paper generalizes the results of the earlier one to the case of scenes that contain polyhedra having arbitrary numbers of planes associated with the vertices. The catalog of the twelve possible pictures of trihedral vertices is shown in Figure 1. When polyhedra with arbitrary numbers of edges incident at each vertex are considered the development of an extended catalog is clearly impossible; it would need to contain an infinite number of entries. What is needed instead is a decision procedure that can be applied to any configuration of labelled lines that is a candidate for inclusion in that hypothetical catalog. We shall also generalize this procedure still further so that it applies not only to the configurations of lines incident at a single picture node, but also to the situation in which an arbitrary number of lines having arbitrary labels enter a