11.1.4 Tolerance region

A tolerance region of a solid is constructed according to the ball-offset operator model as in Rossignac [351]. Farouki [93] studied the problem of finding exact offsets on the exterior of simple solids. He handled the tangent discontinuities at the edge and vertices by using the rolling ball offset definition. The class of solids studied are closed convex solids like solids of revolution and extrusion.

Patrikalakis and Bardis [297] construct a tolerance region of a quadrilateral design surface patch which is bounded by ten surfaces as illustrated in Fig. 11.3. For the shapes of interest in practical applications and the strict tolerance requirements under consideration, they assumed that there will be no self-intersections in the ten bounding surfaces. These ten surfaces are: 1) four pipe or canal surfaces (see Sect. 11.6), the offsets to the edges of the design surface; 2) two offset surfaces, one along the normal and one in the opposite direction to that of the normal to the design surface; 3) four spherical surface patches, the offsets of the corners of the design surface. They approximated the pipe surfaces with rational B-splines and the normal offsets by integral B-splines and expressed the spherical segments exactly by rational B-spline surface patches. It is interesting to note that the design surface is the medial axis of the tolerance region.