Computing view graphs of algebraic surfaces (Q1314433)

Given an embedded smooth surface \(S \subset \mathbb{R}^ 3\), the view space \({\mathcal V}\) is the space of projection directions \(\mathbb{P}^ 2\) or the space of centers of projections \(\mathbb{R}^ 3 \backslash S\). For a given \(\omega \in {\mathcal V}\), the apparent contour of \(S\) is defined as the set of critical values of the projection defined by \(\omega\). The ``bad'' points \(\omega\), for which the apparent contour of \(S\) has singularities worse than cusps and ordinary nodes, define a bifurcation set \({\mathcal B}\) of positive codimension in the view space. The view graph of \(S\), which reflects the ``visual complexity'' of \(S\), is obtained by assigning a node to each connected component of \({\mathcal V} \backslash {\mathcal B}\), and an edge joining two nodes whose corresponding components are separated by a branch of \({\mathcal B}\) of codimension 1. In this paper, the author shows that the view graph of a compact embedded surface given as the zero-set of a polynomial of degree \(d\) has at most \(O (d^{10 \dim {\mathcal V}})\) nodes. He also describes an algorithm for the computation of view graphs in the case of surfaces defined by rational polynomials. Some examples of degree four are explicitly computed.