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each other. Equivalently, we can negate shape B 's vertices and normals and B 's
position relative to A and use s . Another way to look at it is to remember that we
implicitly operate on the Minkowski sum of A
B .
To adjust for this asymmetry, the Gauss map (see Section 4.5 for an introduc-
tion to Gauss maps) of B has to be negated (by negating normals of B )before
superimposing it with the Gauss map A , as in Figure 4.17(a) and (b). All the for-
mulae in this chapter take this negation into account, and both n A and n B denote
nonnegated normals of shapes A and B , correspondingly.
⊕−
4.5
Intuitive Gauss Map
At first encounter, the Gauss map may be a somewhat unintuitive and even intim-
idating concept. It is, however, a widely used tool [Fogel and Halperin 07]. This
section is a short introduction to the Gauss maps of a polytope.
The Gauss map of the surface of a polytope is the surface of the unit sphere
2 . Each point on the polytope surface maps simply to its normal. Hence the
unit sphere: the normals are all unit length. Also, it means that all the points on a
polytope face map to one single normal—also a point on a unit sphere—the face's
normal. (See Figure 4.6(a ). )
Edges don't have definite normals per se, but it's natural to assign them the
range of normals between their adjacent faces. If you imagine the edge as a thin
smooth bevel, it's a natural and intuitive extension. Thus, edges map to great arcs
on
S
S
2 .(See Figure4.6(b). )
Figure 4.6. Gauss map components.