Gauss map computation for free-form surfaces

The Gauss map of a smooth doubly{curved surface characterizes the range of variation of the surface normal as an area on the unit sphere. An algorithm to approximate the Gauss map boundary to any desired accuracy is presented, in the context of a tensor{product polynomial surface patch, r(u;v) for (u; v) 2 0; 1 ] 0; 1 ]. Boundary segments of the Gauss map correspond to variations of the normal along the patch boundary or the parabolic lines (loci of vanishing Gaussian curvature) on the surface. To compute the latter, points of vanishing Gaussian curvature are identiied with the zero{set of a bivariate polynomial, expressed in the numerically{stable Bernstein basis | the subdivision and variation{diminishing properties then govern an adaptive quad-tree decomposition of the (u; v) parameter domain that captures the zero{set of this polynomial to any desired accuracy. Loci on the unit sphere corresponding to the patch boundaries and parabolic lines are trimmed at their mutual or self{intersection points (if any), and the resulting segments are arranged in a graph structure with the segment end{points as nodes. By appropriate traversal of this graph, the Gauss map boundary segments may then be identiied in proper order, and extraneous segments (lying in the Gauss map interior) are discarded. The symmetrization of the Gauss map (by identiication of antipodal points) and its stereographic projection onto a plane are also discussed.