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where
U
is chosen in some appropriate consistent manner as we move from point to
point. (For example, Calladine ([Call86]) choses
U
to have one third of the area of all
the facets adjacent to
q
.)
The validity of an earlier comment is now clear. The polygonal Gauss curvature
vanishes at nonvertex points. If the point lies in the interior of a facet, then the normal
neighborhood is just a point. If the point lies in the interior of an edge, then the normal
neighborhood is just an arc of a great circle. In both cases, the area of the neighbor-
hoods is zero.
Curvature is related to another important quantity in the polygonal context.
Definition.
The
angular defect
at a point
p
in
S
is defined to be 0 if
p
is not a vertex
and if
p
is a vertex, then it is
2p-
(
)
sum of the interior angles at of the facets of that meet at the
p
S
p
.
See Figure 9.22 again. The angular defect at the vertex
p
in Figure 9.22(a) is the
angle a shown in Figure 9.22(b). The angular defect clearly has some relation to cur-
vature, because the larger it is, the more pointed the surface is at a vertex.
Let
p
be a point of a polygonal surface
S
in
R
3
. Then
9.9.4. Theorem.
angular defect at in
pS
=
normal angle of in .
pS
Proof.
The only case that has to be considered is where
p
is a vertex. See Figure
9.22. The angle a in Figure 9.22(b) is the angular defect at the vertex
p
in Figure
9.22(a). The area of the shaded region in Figure 9.22(c) is the normal angle of
p
. See
[Call86], [Crom97], or [HilC99] for a proof. A basic element is the classical Greek
theorem that states that the area of a region in a sphere that is bounded by arcs of
great circles depends only on sum of the exterior angles of the region.
A consequence of Theorem 9.9.4 is worth noting, namely, the normal angle at a
vertex is unchanged if the surface is “folded” arbitrarily at the vertex by deformations
that move the facets rigidly changing the “creases” (the angles between the faces along
a common edge coming out of the vertex). This is because the angular defect does not
change. When facets are triangles and there are more than three at the vertex, then
many different folds are possible. [Call86] describes some applications of this result.
The discussion up to this point in this section was intended to motivate some
important concepts about surfaces and was not entirely rigorous. We shall now con-
sider smooth surfaces again and redo some of what we have covered but will use
definitions that permit a more rigorous development. First of all, returning to the
Gauss map
n
(
p
), note that its derivative D
n
(
p
) at a point
p
of
S
maps the tangent
space T
p
(
S
) to the tangent space to
S
2
at
n
(
p
). Since the two tangent spaces are the
same (the tangent planes are parallel), we consider D
n
(
p
) to be a linear map
()
()
Æ
()
DT
np
:
S
T
S
.
p
p
Definition.
The map D
n
(
p
) is called the
Weingarten map
.
Note.
We have given different definitions of surfaces and manifolds and their
tangent spaces. Therefore, to avoid any confusion on the part of the reader when it
