Polyhedral Scene Analysis Combining Parametric Propagation with Calotte Analysis

Abstract

Polyhedral scene analysis studies whether a 2D line drawing of a 3D polyhedron is realizable in the space, and if so, parameterizing the space of all possible realizations. For generic 2D data, symbolic computation with Grassmann-Cayley algebra is necessary. In this paper we propose a general method, called parametric calotte propagation, to solve the realization and parameterization problems in polyhedral scene analysis at the same time. Starting with the fundamental equations of Sugihara in the form of bivector equations, we can parameterize all the bivectors by introducing new parameters. The realization conditions are implied in the scalar equations satisfied by the new parameters, and can be derived by further analysis of the propagation result. The propagation procedure generally does not bifurcate, and the result often contains equations in factored form, thus makes further algebraic manipulation easier. In application, the method can be used to find linear construction sequences for non-spherical polyhedra.