Notes on the complexity of exact view graph algorithms for piecewise smooth Algebraic Surfaces

Abstract

The view graph of a surface N in 3-space is a graph embedded in the space ν of centers or directions of projection, whose nodes correspond to maximal connected regions of ν which yield equivalent views of N. The size of the view graph of a piecewise smooth algebraic surface N with transverse self-intersection curves and isolated triple-points and cross-caps is O(n kdimν d 6dimν), where n and d denote the number of “component surfaces” of N and their maximal degree, respectively, and where K=6 in general or K=3 for N diffeomorphic to the boundary of a polyhedron. (For surfaces without cross-caps, this bound has been established in [17].) Also, for the special piecewise linear case, where d=1 and K=3, it is known that the size of the view graph is actually Θ(n 3dimν). It is shown that the exact view graphs of such surfaces can be determined in O(n K(2dimν+1)). · P(d, L) time by a deterministic algorithm and in O(n Kdimν+e) · P(d, L) expected time by a randomized algorithm. Here P is some polynomial, L is the maximal coefficient size of the defining polynomials of N, and e is an arbitrarily small positive constant. Note that the randomized algorithm is, in terms of combinatorial complexity (where d and L are assumed to be constants which do not depend on n), nearly optimal—its combinatorial time complexity exceeds the size of the view graph only by e in the exponent.