420 20 Geodetic mapping
Box 20.1 (Surface geometry of E
2
A 1 ,A 2 ).
Matrices
`
Frobenius matrix F (elements a, b, c, d), Gauss matrix G = J
T J, (elements e, f , g),
Hesse matrix H =
˘˙
X , KL , G 3
¸¯
(elements l, m, n), curvature matrix K,
Jacobi matrix J =
˘
∂X
J /∂U
K
¯´
:
F = {F
1
KL } =
2
6
4
√
1−E 2 sin 2 B
A 1 cos B
0
0
(1−E
2 sin
2 B)
3/2
A 1 (1−E 2 )
3
7
5 , F ∈ R
2×2 ,
(20.13)
G = {G
1
KL } =
2
4
A
2
1 cos
2 B
1−E 2 sin 2 B
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 B) 3
3
5 , G ∈ R
2×2 , G = J
T J ,
(20.14)
H = {H
1
KL } =
2
6
4
−
A 1 cos
2 B
√
1−E 2 sin 2 B
0
0
−
A 1 (1−E
2 )
(1−E 2 sin 2 B) 3/2
3
7
5 , H ∈ R
2×2 ,
(20.15)
K = {K
1
KL } =
2
6
4
√
1−E 2 sin 2 B
A 1
0
0
(1−E
2 sin
2 B)
3/2
A 1 (1−E 2 )
3
7
5 = −HG
−1 , K ∈ R
2×2 ,
(20.16)
h = −
tr[K]
2
= −
√
1−E 2 sin 2 B(2−E
2 (1+sin
2 B))
2A 1 (1−E 2 )
,
(20.17)
k = det[K] =
(1−E
2 sin
2 B)
2
A 2
1
(1−E 2 )
,
(20.18)
J = {J
1
KL } =
∂(X,Y )
∂(L,B)
=
2
6
6
6
6
6
4
−
A 1 cos B sin L
√
1−E 2 sin 2 B
−
A 1 (1−E
2 ) sin B cos L
(1−E 2 sin 2 B) 3/2
A 1 cos B cos L
√
1−E 2 sin 2 B
−
A 1 (1−E
2 ) sin B sin L
(1−E 2 sin 2 B) 3/2
0
+
A 1 (1−E
2 ) cos B
(1−E 2 sin 2 B) 3/2
3
7
7
7
7
7
5
, J ∈ R
3×2 .
(20.19)
Eigenvalues:
1st eigenvalue of K: κ 1 =
p
1 − E 2 sin
2 B/A 1 ;
κ
−1
1 =: N (B) = A 1 /
p
1 − E 2 sin
2 B (1st curvature radius);
2nd eigenvalue of K: κ 2 = (1 − E
2 sin
2 B)
3/2 /A 1 (1 − E
2 );
κ
−1
2 =: M (B) = A 1 (1 − E
2 )/(1 − E
2 sin
2 B)
3/2 (2nd curvature radius).
Christoffel symbols
˘ M
KL
¯
:
{
1
11 }(L, B) = {
1
22 }(L, B) = {
2
12 }(L, B) = 0 , {
1
12 }(L, B) = −
− tan B(1−E
2 )
1−E 2 sin 2 B
,
{
2
11 }(L, B) =
sin B cos B(1−E
2 sin
2 B)
1−E 2
, {
2
22 }(L, B) = 3E
2 sin B cos B(1 − E
2 sin
2 B) .
(20.20)