Graphics Reference
In-Depth Information
Figure 8.16.
Continuously varying
choice of orientation.
F(q
1
)
R
k
F(q
2
)
V
M
F
U
q
1
q
2
R
n
Exercise 8.5.3 describes another way to define the orientability of a manifold.
Definition.
Let F :
U
Æ
M
k
be a regular parameterization of a k-dimensional mani-
fold
M
. The orientation s of
M
that associates to each
p
=F(
q
) Œ
M
the orientation
∂
∂
FF
()
∂
∂
∂
∂
F
È
Í
˘
˙
()
()
q
,
q
,...,
q
(8.9)
u
u
u
k
1
2
of the tangent space T
p
(
M
) is called the
standard orientation of
M
induced by
F.
Dealing with ordered bases is not very convenient and so we now describe a better
way to specify an orientation of a manifold in a common special case, but first some
definitions.
Definition.
Let
M
k
be a k-dimensional C
r
submanifold of
R
n
. A
C
r
vector field
of
R
n
defined over
M
is a C
r
(vector-valued) function
k
n
n:
MR
Æ
.
The vector field n is called
tangential to
M
or simply a
C
r
vector field of
M
if n(
p
) Œ
T
p
(
M
) for all
p
Œ
M
. The vector field n of
R
n
is called
normal to
M
or a
C
r
normal
vector field of
M
in
R
n
if n(
p
) is orthogonal to T
p
(
M
) for all
p
Œ
M
. (The phrase “in
R
n
” is often dropped if
R
n
is clear from the context.) In any case we say that the vector
field is a
unit vector field
or a
nonzero vector field
if n(
p
) has unit length or is nonzero,
respectively, for all
p
Œ
M
.
Vector fields of manifolds associate vectors to points of a manifold with the vector
at a point lying in the tangent space (or plane) at that point. Figure 8.17(a) shows a
vector field of
S
1
. Figure 8.17(b) shows a normal vector field of
S
1
in
R
2
. As usual, the
adjective “C
r
” will be suppressed. The two typical cases are continuous (C
0
) or C
•
.
Nonzero vector fields are often normalized to unit vector fields.
