Graphics Reference
In-Depth Information
Shape operator:
S
p
=-D
n
(
p
): T
p
(
S
) Æ T
p
(
S
)
Normal curvature of g at
p
in
S
:
k
n,g
(
p
) = k
N
•
n
(
p
),
where
N
is the principal normal and k is the
curvature to g at
p
.
Choose orthonormal basis (
u
1
,
u
2
) for T
p
(
S
) so that
D
n
(
p
)(
u
1
) =-k
1
u
1
(S
p
(
u
1
) = k
1
u
1
)
D
n
(
p
)(
u
2
) =-k
2
u
2
(S
p
(
u
2
) = k
2
u
2
).
and assume that k
1
≥ k
2
.
Principal normal curvatures:
k
1
, k
2
(The maximum and minimum of Q
II
on the unit circle of T
p
(
S
))
Principal normal directions:
u
1
,
u
2
Gauss curvature:
K(
p
) = determinant of D
n
(
p
)
LN
-
-
M
2
K = k
1
k
2
=
2
EG
F
1
2
()
=-
(
()
)
Mean curvature:
H
p
Trace d
n p
kk
1
+
EN
+-
-
GL
2
FM
2
H
=
=
(
)
2
2
2
EG
F
MF
-
-
LG
LF
-
-
ME
Weingarten equations:
n
u
=
F
+
F
u
v
2
2
EG
F
EG
F
NF
-
-
MG
MF
-
-
NE
n
v
=
F
+
F
u
v
2
2
EG
F
EG
F
k
2
Principal normal curvatures are roots of:
- 2Hk + K = 0
2
2
k
=+
HHK
-
,
k
=-
HHK
-
1
2
G
ij
Christoffel symbols:
GE
-
2
FF
+
FE
GE
-
-
FG
2
GF
-
GG
-
FG
u
u
v
v
u
v
u
v
1
1
1
1
G
=
,
G
==
G
,
G
=
,
11
(
)
12
21
(
)
22
(
)
2
2
2
2
EG
-
F
2
EG
F
2
EG
-
F
2
EF
-
EE
+
FE
EG
-
-
FE
EG
-
2
FF
+
FFG
u
v
u
u
v
v
v
u
1
2
1
2
2
2
2
G
=
,
G
==
G
,
G
=
.
(
)
21
(
)
(
)
2
2
2
2
EG
-
F
2
EG
F
2
EG
-
F
