Geography Reference
In-Depth Information
From these mapping equations, we derive
cha-cha-cha
(“change from one chart to another chart”)
tan
V
cos
U
tan
B
cos
L
,
tan
L
=
−
versus tan
U
=
−
(20.7)
tan
U
tan
L
1+tan
2
V/
cos
2
U
1+tan
2
B/
cos
2
L
tan
B
=
tan
V
=
versus
(20.8)
sin
U
√
cos
2
U
+tan
2
V
=
sin
L
=
√
cos
2
L
+tan
2
B
,
in particular, the diffeomorphism
d
U
d
V
=
=
d
L
d
B
,
sin
L
tan
B
cos
2
L
+tan
2
B
cos
L
cos
2
B
cos
2
L
+sin
2
B
−
−
(20.9)
cos
L
(cos
2
L
+tan
2
B
)
1
/
2
sin
L
tan
B
(cos
2
L
+tan
2
B
)
1
/
2
−
d
L
d
B
=
=
−
tan
V
sin
U
d
U
d
V
.
−
cos
U
cos
2
V
(cos
2
U
+tan
2
V
)
cos
2
U
+tan
2
V
(20.10)
cos
U
(cos
2
U
+tan
2
V
)
1
/
2
sin
U
tan
V
(cos
2
U
+tan
2
V
)
1
/
2
−
2
3
,δ
IJ
Boxes
20.1
and
20.2
summarize the surface geometry of
, in particular, the
matrices F, G, H, and J of type Frobenius, Gauß, Hesse, and Jacobi as well as the curvature
matrix K :=
E
A
1
,A
2
⊂{
R
}
HG
−
1
, especially the
surface fundamental forms
−
{
I
,
II
,
III
}
.
A
1
,A
2
be a smooth curve which is parameterized by arc length.
Denote by
{
D
1
,
D
2
,
D
3
}
its
Darboux frame
defined by (
20.11
). Its derivational equations are
given by (
20.12
), introducing the
antisymmetric connection matrix
Ω(
κ
g
,κ
n
,τ
g
) containing
geode-
tic curvature κ
g
,
normal curvature κ
n
,and
geodetic torsion τ
g
. In terms of the first fundamental
form, in particular
{G
KL
}
, the second fundamental form, in particular
{H
KL
}
, the Riemann con-
Let
C
:[0
,
∞
]
→{
E
,G
KL
}
nection, in particular the Christoffel symbols
M
, the curvature measures of the curve
C
as
KL
A
1
,A
2
a submanifold of
{
E
,G
KL
}
can be represented by Corollary
20.1
.
U
k
(
S
)
D
1
:=
d
X
{
}
=
∂
X
∂U
K
U
K
=
G
K
U
K
,
d
S
D
2
:=
∗
(
D
3
∧
D
1
)=
D
3
×
D
1
,
(20.11)
D
3
:=
G
3
,
D
1
=+
κ
g
D
2
+
κ
n
D
3
,
D
2
=
−
κ
g
D
1
+
τ
g
D
3
,
D
3
=
−
κ
n
D
1
−
τ
g
D
2
,
(20.12)
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
D
1
D
2
D
3
0
κ
g
κ
n
D
1
D
2
D
3
0
κ
g
κ
n
−
⎣
⎦
=
⎣
⎦
⎣
⎦
,
D
=
Ω
D
,
Ω
=
⎣
⎦
.
−
κ
g
0
τ
g
−κ
n
−τ
g
0
κ
g
0
τ
g
−κ
n
−τ
g
0
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