On the complexity and computation of view graphs of piecewise smooth algebraic surfaces

Abstract

Families of central and parallel projections of piecewise smooth surfaces (i.e. surfaces whose singular set consists of curves of transverse double points and isolated triple points) onto planes are studied. It is shown that the bifurcation set in the parameter space of such a family has positive codimension for any piecewise smooth C∞-surface. For the special case of piecewise smooth algebraic surfaces M , which consist of n non-singular real algebraic surfaces of degree not more than d intersecting along curves of transverse double points and in isolated triple points, the number of connected components in the complement of the bifurcation set is bounded by a polynomial in n and d . A symbolic algorithm that determines exactly one sample point (with rational coordinates) in each connected component is presented, these sample points are the nodes in the view graph of M .