Graphics Reference
In-Depth Information
Figure 15.1.
Curve in parameterized
surface.
(
)
=
(
(
)
(
)
(
)
)
j
uv
,
j
uv
,
,
j
uv
,
,
j
uv
,
1
2
3
and
()
=
(
()
()
)
aaa
t
t
,
t
.
1
2
We know that
∂
∂
jjjj
∂
∂
∂
∂
∂
∂
∂
∂
jjjj
∂
∂
∂
∂
∂
∂
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
1
2
3
1
2
3
=
,
,
and
=
,
,
u
u
u
u
v
v
v
v
form a basis for the tangent space at every point of the surface and
∂
∂
jj
uv
∂
∂
n
=
¥
is a normal vector at those points. The chain rule implies that
¢
()
=
(
¢
()
)
g
tD
j a
t
T
=
()
(
)
aj
tJ uv
,
∂
∂
j
∂
∂
j
∂
∂
j
Ê
ˆ
1
2
3
(
uv
,
)
(
uv
,
)
(
uv
,
)
Á
Á
Á
˜
˜
˜
u
u
u
=¢
()
(
aa
t
¢
()
t
)
(15.1)
1
1
∂
∂
j
∂
∂
j
∂
∂
j
1
2
3
(
)
(
)
(
)
uv
,
uv
,
uv
,
Ë
¯
v
v
v
where Jj is the Jacobian matrix for j. It follows that
T
T
¢
(
¢
gajaj
≤=
≤
J
+
J
(15.2)
and
