Offset Curves and Surfaces

Abstract

Offset curves/surfaces, also called parallel curves/surfaces, are defined as the locus of the points which are at constant distant d along the normal from the generator curves/surfaces. A literature survey on offset curves and surfaces was carried out by Pham [313] and more recently by Maekawa [250]. Offsets are widely used in various applications, such as tool path generation for **** pocket machining [157, 153, 349], 3-D NC machining [116, 52, 215, 366] (se Fig. 11.1), in feature recognition through construction of skeletons or medial axes of geometric models [298, 450] (see Fig. 11.2), definition of tolerance regions [93, 353, 297] (see Fig. 11.3), access space representation in robotics [237] (see Fig. 11.4), curved plate (shell) representation in solid modeling [301] (see Fig. 11.5), rapid prototyping where materials are solidified in successive two-dimensional layers [114] and brush stroke representation [198]. Because of the square root involved in the expression of the unit normal vector, offset curves and surfaces are functionally more complex than their progenitors. If the progenitor is a rational B-spline, then its offset is usually not a rational B-spline, except for special cases including cyclide surface patches [332, 83, 404], Pythagorean hodograph curves and surfaces (see Sect. 11.4) and simple solids [93]. Another difficulty arises when the progenitor has a tangent discontinuity. The its exterior and interior offset will become discontinuous or have self-intersections as illustrated in Fig. 11.6. Further-more offsets may have cups and self-intersection, even if the progenitor is regular (see Figs. 11.9, 11.25). Frequently in applications, discontinuity in offsets must be filled in and the loops arising from self-intersections must be trimmed off. In the following three sections, we will briefly review some of the literature on NC machining, medial axis transforms and tolerance regions.