Graphics Reference
In-Depth Information
9.19
Summary of Surface Formulas
In the formulas below F(u,v) is a parameterization of a surface
S
in
R
3
and
∂
∂
F
∂
∂
F
F
=
and
F
=
.
u
v
u
v
g(t) will be a regular curve on
S
and assumed to be expressed in the form g(t) =F(m(t)),
where m(t) = (u(t),v(t)) is a curve in
R
2
.
First fundamental form:
A quadratic form Q
I
defined on the tangent
space T
p
(
S
) by
Q
I
(
v
) =
v
•
v
Metric coefficients:
Q
I
(g¢) = E(u¢)
2
+ 2F(u¢)(v¢) + G(v¢)
2
E =F
u
• F
u
, F =F
u
• F
v
, G =F
v
• F
v
|F
u
¥F
v
|
2
For first fundamental form:
= EG - F
2
∂
∂
FF
∂
∂
For a general parameterization F(u
1
,u
2
,...,u
n
):
g
=∑
ij
uu
i
j
t
()
=
Ú
2
2
Length
Eu
¢
+
2
Fu v
¢
¢ +
Gv
¢
Length:
0
FF
FF
∑
F
EG
u
v
uv
Angle a between F
u
and F
v
:
cos
a
=
=
Ú
2
Area of
S
with
U
the domain of F:
EG
-
F
U
Volume of a parameterization F with domain
U
(u
1
,u
i
,...,u
n
) for a manifold
M
n
in
R
n+1
:
Ú
Ú
()
=
()
V
det g
ij
U
()
=
g
,
where g
=
det
g
ij
U
n:S
Æ
S
2
, where
n
(
p
) is the unit normal at
p
for the oriented surface
S
Gauss map:
Second fundamental form:
A quadratic form Q
II
defined on the tangent
space T
p
(
S
) by
Q
II
(
v
) =-D
n
(
p
)(
v
)•
v
Q
II
(v) = L(u¢)
2
+ 2M(u¢)(v¢) + N(v¢)
2
L =-
n
u
• F
u
=
n
• F
uu
,
M =-
n
v
• F
u
=
n
• F
uv
=
n
• F
vu
=-
n
u
• F
v
N =-
n
v
• F
v
=
n
• F
vv
.
Coefficients of second fundamental form:
