Graphics Reference
In-Depth Information
9.8.2. Example.
To compute E, F, G for the parameterization F(u,v) =
p
0
+ u
w
1
+ v
w
2
, (u,v) Œ
R
2
, of the plane through the point
p
0
with orthonormal basis
w
1
and
w
2
.
Solution.
Now
∂
∂
F
∂
∂
F
=
w
and
=
w
1
2
u
v
implies that
g
=
1
,
g
=
g
=
0
,
and
g
=
1
.
11
12
21
22
Therefore, E = 1, F = 0, and G = 1.
9.8.3. Example.
To compute E, F, G for the parameterization
(
)
=
(
)
(
)
Œ
[
]
¥
F uv
,
cos u, sin u, v
,
uv
,
0 2p
,
R
,
of the cylinder x
2
+ y
2
= 1.
Solution.
This time
∂
∂
F
∂
∂
F
(
)
=
(
)
=-
sin u, cos u,0
and
0 0 1
,, ,
u
v
so that
g
=
1
,
g
=
g
=
0
,
and
g
=
1
.
11
12
21
22
Therefore, E = 1, F = 0, and G = 1. It may seem a little strange that the metric coeffi-
cients of the cyclinder are the same as those of a plane, but we shall see why this is
as it should be shortly.
9.8.4. Example.
To compute E, F, G for the spherical coordinate parameterization
)
Œ
[
]
¥
[
]
(
)
=
(
)
(
Fqf
,
r cos
q
sin , r sin
f
q
sin , r cos
f
f
,
qf
,
0 2
,
p
0
,
p
,
of the sphere of radius r about the origin.
Solution.
We have
∂
∂q
F
∂
∂f
F
(
)
(
)
=-
r
sin
qf
sin , r cos
qf
sin , 0
and
=
r
cos
qf
cos , r sin
qf
c
os
,
-
r sin
f
.
Therefore, E = r
2
sin
2
f, F = 0, G = r
2
.