Graphics Reference
In-Depth Information
between directional derivatives and differentials (equation (4.31b) in Chapter 4), it
follows that w
12
= dq. This implies that w
21
=-dq. It is easy to see that all other w's
are zero.
To understand the motion of frame fields better, one introduces dual forms.
Given a frame field (
U
1
,
U
2
,
U
3
) on
R
3
, define the
dual forms
q
i
(
p
) Œ
Definition.
T
p
(
R
3
)* by
()
( ()
=∑
()
3
q
i
pv
v Up
,
v
Œ
T
R
.
i
p
As usual, q
i
(
p
) will be abbreviated to q
i
unless the point
p
needs to be specified
explicitly.
9.16.9. Theorem.
(The Cartan structural equations) Let (
U
1
,
U
2
,
U
3
) be a frame field
on
R
3
. Let w
ij
and q
i
be the connection and dual forms, respectively, of this frame field.
Then
3
Â
1
(1) (The first structural equations)
d
i
q
=
wq
Ÿ
ij
j
j
=
3
Â
1
(2) (The second structural equations)
d
w
=
w
Ÿ
w
ij
ik
kj
k
=
Proof.
See [ONei66].
Next, we move on to surfaces. Just like in the case of curves in
R
3
, the geometry
of a surface can be deduced by analyzing special frame fields on them.
Let
S
be a surface in
R
3
.
Definition.
A frame field (
U
1
,
U
2
,
U
3
) on
S
is called an
adapted frame field
on
S
if
U
3
is a normal vector field for
S
.
Again, everything we do will be local in nature, so that we do not need adapted
frame fields to be defined on all of
S
, only on an open subset that is relevant at the
time. Because adapted frame fields are somewhat special, it is useful to make clear
when they exist.
9.16.10. Proposition.
A surface
S
admits an adapted frame field if and only if it is
orientable and has a nonzero (tangent) vector field.
Proof.
Easy.
9.16.11. Example.
The cylinder defined by the equation
2
2
2
xy r
+=
admits the adapted frame field (
U
1
,
U
2
,
U
3
) defined by
