Optimal slicing of free-form surfaces

Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser scanner or coordinate measuring machine, involve sampling of free-from surfaces along section cuts by a family of parallel planes with equidistant spacing Δ and common normal N. To ensure that such planar sections provide faithful descriptions of the shape of a surface, it is desirable to choose the relative orientation that maximizes, over the entire surface, the minimum angle between N and the local surface normal n. We address this optimization problem by computing the (symmetrized) Gauss map for the surface, projecting it stereographically onto a plane, and invoking the medial axis transform for the complement of its image to identify the orientation N that is "most distant" from the symmetrized Gauss map boundary. Using a Gauss map algorithm described elsewhere, the method is implemented in the context of bicubic Bézier surfaces, and applied to the problem of minimizing the greatest scallop height incurred in contour machining of surfaces using a 3-axis milling machine with a ball-end cutter.