Graphics Reference
In-Depth Information
comes to computations, we again restate our assumption that in this chapter we are
following the definitions in Section 8.4. Tangent spaces are considered to be defined
in terms of tangent vectors to curves and derivatives map tangent vectors to tangent
vectors of the composite curves. For example, in case of the Gauss map, if
v
is a vector
in T
p
(
S
) and g(t) is a curve in
S
with g(0) =
p
and g¢(0) =
v
, then
)
¢
()
D
np v
( ()
=
(
n
o
g
0.
(9.43)
9.9.5. Lemma.
The Weingarten map D
n
(
p
) is a self-adjoint linear map.
Proof.
One needs to show that
( ( )
∑=∑
( ( )
D
np v
v
v
D
np v
1
2
1
2
for an arbitrary basis (
v
1
,
v
2
) of T
p
(
S
). This involves fairly straightforward computa-
tions using the definitions. See [DoCa76].
Lemma 9.9.5 is a technical fact, which, along with the next two definitions, will
lead to some geometric results.
Definition.
The quadratic form Q
II
on T
p
(
S
) defined by
Q
II
v
()
=-
D
n p v
( ()
∑
v
is called the
second fundamental form
of
S
at
p
.
The minus sign is used to reduce number of minus signs elsewhere. The second
fundamental form is really a directional derivative. It measures the turning of the
tangent plane in the direction defined by a tangent vector.
Definition.
The map
()
()
Æ
()
SD
=-
np
:
T
S
T
S
p
p
p
is called the
shape operator
for the surface
S
.
Some authors (see [ONei66]) make the latter the basis of the study of surfaces. It
clearly contains the same information as the second fundamental form. If the point
p
is arbitrary, we shall drop the subscript
p
in the notation for the shape operator and
simply write S (in the same sense that one could simply write D
n
). For example, the
notation S(
v
)•
v
is an abbreviation for the statements S
p
(
v
)•
v
for
p
Œ
S
with
v
the
appropriate vector in T
p
(
S
).
Definition.
Let g(t) be a regular curve in
S
passing through
p
. Let
N
be the princi-
pal normal and the k curvature of g at
p
. Let q be the angle between
N
and
n
at
p
.
The number
()
=
()
k
p
k
cos
q
=
k
N
∑
n p
n,
g
is called the
normal curvature
of g at
p
in
S
.
We analyze this definition of normal curvature further. Let g(s) be a (regular) curve
lying in the surface
S
parameterized by arc-length and let
N
(s) be its principal normal.
