Graphics Reference
In-Depth Information
Figure 9.21.
The Gauss map.
N
p
S
n
z
N
S
2
y
x
Gauss who, in 1827, presented the first truly intrinsic surface invariants. Let
S
be an
oriented
surface in
R
3
. Recall that saying
S
is oriented is equivalent to saying that we
have made a consistent choice of unit normals at each point of
S
so that the map
2
nS S
:
Æ
,
where
n
(
p
) is the chosen unit normal at
p
in
S
, is a nice smooth map.
Definition.
The map
n
is called the
Gauss map
for the
oriented
surface
S
.
See Figure 9.21. Compare this definition with what we called the Gauss map
for plane curves in Section 9.3. The assumption that the surface is oriented is
never a problem. First of all, we usually only need it for a small neighborhood of
a point. Second of all, we will be dealing with parameterized surfaces in which
case we can, and shall, always assume that the orientation is induced by the given
parameterization.
Following Gauss we can now define a fundamentally important intrinsic invari-
ant of surfaces.
A geometric definition of Gauss curvature
.
Let
n
be the Gauss map for some
surface
S
. Then the value K(
p
) defined by
(
()
)
lim
“
signed area
area
”
nU
()
=
K
p
,
(9.42)
()
U
Up
Æ
where
U
is a neighborhood of
p
in
S
, is called the
Gauss curvature
of
S
at
p
. By the term
“signed area” in the numerator we mean that we set it to the positive or negative value
of area(
n
(
U
)) depending on whether the map
n
is orientation preserving or not.
This definition is not very rigorous, although it could be made so. We shall give a
better definition shortly, but it is a nice intuitive way to think about the concept.
