Game Development Reference

In-Depth Information

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.