A correspondence between scene analysis and motions of frameworks

Abstract

For over 100 years there has been a known correspondence between plane pictures of spherical polyhedra and static stresses in bar and joint frameworks with planar graphs. We define a new map from any plane picture of a general scene to a corresponding plate and bar framework in the plane. This map, and its return map, give a simple isomorphism between the space of scenes over the picture and the space of instantaneous motions of the framework. When two simple equivalence relations on the pictures and on the frameworks are introduced, the maps define an isomorphism of the equivalence clases which permits the full translation of questions, techniques and theorems between the two fields of study. A direct geometric interpretation of this correspondence is described and an extension to systems with occlusion and tensegrity (systems of inequalities) is presented. The analogous patterns in higher dimensions are described, yielding a correspondence between motions of bar and body frameworks in 3-space and 6-dimensional scenes over 5-dimensional pictures.