Graphics Reference
In-Depth Information
1
(
)
=
(
)
¥
(
)
=-
(
(
)
+
(
)
)
U
xyz
,,
U
xyz
,,
U
xyz
,,
y
E
xyz
,,
x
E
xyz
,,
1
2
3
1
2
r
(
)
=
(
)
U
xyz
,,
E
xyz
,,
2
3
1
(
)
=
(
(
)
+
(
)
)
U
xyz
,,
x
E
xyz
,,
y
E
xyz
,, .
3
1
2
r
9.16.12. Example.
Consider the sphere defined by the equation
2
2
2
2
xyz r
++=.
Because the sphere does not have any nonzero vector field (Corollary 8.5.6) there is
no adapted frame field defined over all of it. On the other hand we can, for example,
define an adapted frame field (
U
1
,
U
2
,
U
3
) over the sphere minus the north and south
pole by defining
1
(
)
=
(
)
=-
(
(
)
+
(
)
)
U
xyz
,,
-
yx
,,
0
y
E
xyz
,,
x
E
xyz
,,
1
1
2
2
2
xy
+
(
)
=
(
)
¥
(
)
U
xyz
,,
U
xyz
,,
U
xyz
,,
2
3
1
1
(
)
=
(
(
)
+
(
)
+
(
)
)
U
xyz
,,
x
E
zyz
,,
y
E
xyz
,,
z
E
x
,
yyz
,
.
3
1
2
3
r
If the sphere were the earth, then the vector
U
1
would point due “east.” See Figure 9.37.
Next, assume that (
U
1
,
U
2
,
U
3
) is an adapted frame field on a surface
S
. Although
the frame field is only defined at points
p
in the surface
S
, as long as we stick to vectors
v
in T
p
(
S
), the connection equation
3
Â
( () ()
—=
v
U
w
p v U p
.
i
ij
j
j
=
1
will still be defined. This follows again because of the way that the directional deriv-
ative is defined—we differentiate in the tangent direction of a curve in the surface
U
2
U
3
U
1
The adaptive frame field on
S
2
in
Example 9.16.12.
Figure 9.37.
