Efficient One-Sided Linearization of Spline Geometry

This paper surveys a new, computationally efficient technique for linearizing curved spline geometry, bounding such geometry from one side and constructing curved spline geometry that stays to one side of a barrier or inside a given channel. Combined with a narrow error bound, these reapproximations tightly couple linear and nonlinear representations and allow them to be substituted when reasoning about the other. For example, a subdividable linear efficient variety enclosure (sleve, pronounced like Steve) of a composite spline surface is a pair of matched triangulations that sandwich a surface and may be used for interference checks. The average of the sleve components, the mid-structure, is a good max-norm linearization and, similar to a control polytope, has a welldefined, associated curved geometry representation. Finally, the ability to fit paths through given channels or keep surfaces near but outside forbidden regions, allows extending many techniques of linear computational geometry to the curved, nonlinear realm.