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For surfaces, such as Bezier or B-splines, or NURBS with positive weights, which are defined by networks of control points and for which identifiable pieces of surface are known to lie within the convex hull of a subset of the control points, recursive division is one of the most robust interrogation techniques available, guaranteeing except in very singular circumstances to find all components of the intersection. Offset surfaces, used where an object has a small non-zero thickness, and to determine cutter center loci where a cutting point must lie on a given surface, have not had this option. This paper describes a technique for applying recursive division interrogation to offset surfaces, where the offset is either constant or a function of surface normal. Sections I and II recapitulate the standard theory of recursive subdivision interrogation and the definition of offset surfaces. Section III explores the concept of procedural interface, and section IV introduces that of a Quantized Hull. Sections V to VII suggest a naive method of recursive division interrogation, discover why it does not work, and show how it may be salvaged.
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