Global bifurcation sets and stable projections of nonsingular algebraic surfaces

摘要

The view graph of a surface is a planar graph whose nodes are the stable views (projections) of the surface and whose edges represent transitional views of codimension one. The space of all directions of orthogonal projection can be identified with the projective plane. The set of “bad” projection directions, associated with the degenerate views of positive codimension, forms a graph in the projective plane (the view bifurcation set). This graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph. We assume that the projected surface is nonsingular and parameterized by polynomials of degree d. We present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization.