Offset curves/surfaces, also called parallel curves/surfaces, are defined as the locus of the points which are at
constant distant
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
-D pocket machining
[157,153,349], 3-D NC machining
[116,52,215,366] (see 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 representations 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. Then its exterior and interior offsets will
become discontinuous or have self-intersections as illustrated in
Fig.
11.6. Furthermore offsets may have cusps and
self-intersections, 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.
Figure 11.1:
NC machining: (a)
-D pocket milling (adapted from [254]), (b) 3-D milling (adapted from [223])
Figure 11.2:
Medial axis (adapted from [139])
Figure 11.3:
Definition of tolerance regions.
Figure 11.4:
Access space representation in robotics (adapted from [237])
Figure 11.5:
Plate representation
Figure 11.6:
Offsets to a tangent
discontinuous curve (adapted from [254])
Next: 11.1.2 NC machining
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December 2009