222 8 “Ellipsoid-of-revolution to tangential plane”
Continuation of Box.
I = I 2 =
»
1 0
0 1
–
∈ R
2×2 .
(8.7)
1st curvature radius, normal curvature:
κ 1 =
1
A 1
p
1 − E 2 sin
2 Φ , κ
−1
1 =
A 1
p
1 − E 2 sin
2 Φ
=: N (Φ) .
(8.8)
2nd curvature radius, meridianal curvature:
κ 2 =
(1 − E
2 sin
2 Φ)
3/2
A 1 (1 − E 2 )
, κ
−1
2 =
A 1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 3/2 =: M (Φ) .
(8.9)
Mean curvature, Gauss curvature:
h = −
p
1 − E 2 sin
2 Φ
2A 1
−
(1 − E
2 sin
2 Φ)
3/2
2A 1 (1 − E 2 )
= −
1
2
N + M
NM
, k =
(1 − E
2 sin
2 Φ)
2
A
2
1 (1 − E 2 )
=
1
MN
. (8.10)
Christoffel symbols of the 2nd kind:
j
1
1 1
ff
=
j
1
2 2
ff
=
j
2
1 2
ff
= 0 ,
j
1
1 2
ff
= −
(1 − E
2 ) tan Φ
1 − E 2 sin
2 Φ
,
j
2
1 1
ff
=
1
2
sin 2Φ
1 − E
2 sin
2 Φ
1 − E 2
,
j
2
2 2
ff
= 3E
2 sin Φ cos Φ(1 − E
2 sin
2 Φ) ;
j
M
K L
ff
:=
1
2
G
MN `
D K G NL + D L G KN − D N G KL
´ ∈ R
2×2×2 ∀ K, L, M ∈ {1, 2} .
(8.11)
In addition to this ID card, we here present the central characteristics of the geodetic ellipsoidal system:
see Table 8.1, “Geodetic Reference System 1980” (Bulletin Geodesique, 58, pp. 388–398, 1984) versus
“World Geodetic Datum 2000” (Journal of Geodesy, 73, pp. 611–623, 1999).
Table 8.1. “Geodetic Reference System 1980” versus “World Geodetic Datum 2000”.
H. Moritz (1984)
E. Grafarend, A. Ardalan (1999)
(“zero frequency tide geoid”)
Semi-major axis
A 1
6 378 137 m
6 378 136.602 ± 0.053 m
Semi-minor axis
A 2
6 356 752.3141 m
6 356 751.860 ± 0.052 m
Relative eccentricity
E
2 = (A
2
1 − A
2
2 )/A
2
1
0.006 694 380 022 90
0.006 694 397 984 91
Absolute eccentricity
=
p
A
2
1 − A
2
2
521 854.0097 m
521 854.674 ± 0.015 m
Axis difference
A 1 − A 2
21 384.686 m
21 384.742 m
Flattening
F = (A 1 − A 2 )/A 1
0.003 352 810 681 18
0.003 352 819 692 40
Inverse flattening
F
−1 = A 1 /(A 1 − A 2 )
298.257 222 101
298.256 420 489
Continuation of Box.
I = I 2 =
»
1 0
0 1
–
∈ R
2×2 .
(8.7)
1st curvature radius, normal curvature:
κ 1 =
1
A 1
p
1 − E 2 sin
2 Φ , κ
−1
1 =
A 1
p
1 − E 2 sin
2 Φ
=: N (Φ) .
(8.8)
2nd curvature radius, meridianal curvature:
κ 2 =
(1 − E
2 sin
2 Φ)
3/2
A 1 (1 − E 2 )
, κ
−1
2 =
A 1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 3/2 =: M (Φ) .
(8.9)
Mean curvature, Gauss curvature:
h = −
p
1 − E 2 sin
2 Φ
2A 1
−
(1 − E
2 sin
2 Φ)
3/2
2A 1 (1 − E 2 )
= −
1
2
N + M
NM
, k =
(1 − E
2 sin
2 Φ)
2
A
2
1 (1 − E 2 )
=
1
MN
. (8.10)
Christoffel symbols of the 2nd kind:
j
1
1 1
ff
=
j
1
2 2
ff
=
j
2
1 2
ff
= 0 ,
j
1
1 2
ff
= −
(1 − E
2 ) tan Φ
1 − E 2 sin
2 Φ
,
j
2
1 1
ff
=
1
2
sin 2Φ
1 − E
2 sin
2 Φ
1 − E 2
,
j
2
2 2
ff
= 3E
2 sin Φ cos Φ(1 − E
2 sin
2 Φ) ;
j
M
K L
ff
:=
1
2
G
MN `
D K G NL + D L G KN − D N G KL
´ ∈ R
2×2×2 ∀ K, L, M ∈ {1, 2} .
(8.11)
In addition to this ID card, we here present the central characteristics of the geodetic ellipsoidal system:
see Table 8.1, “Geodetic Reference System 1980” (Bulletin Geodesique, 58, pp. 388–398, 1984) versus
“World Geodetic Datum 2000” (Journal of Geodesy, 73, pp. 611–623, 1999).
Table 8.1. “Geodetic Reference System 1980” versus “World Geodetic Datum 2000”.
H. Moritz (1984)
E. Grafarend, A. Ardalan (1999)
(“zero frequency tide geoid”)
Semi-major axis
A 1
6 378 137 m
6 378 136.602 ± 0.053 m
Semi-minor axis
A 2
6 356 752.3141 m
6 356 751.860 ± 0.052 m
Relative eccentricity
E
2 = (A
2
1 − A
2
2 )/A
2
1
0.006 694 380 022 90
0.006 694 397 984 91
Absolute eccentricity
=
p
A
2
1 − A
2
2
521 854.0097 m
521 854.674 ± 0.015 m
Axis difference
A 1 − A 2
21 384.686 m
21 384.742 m
Flattening
F = (A 1 − A 2 )/A 1
0.003 352 810 681 18
0.003 352 819 692 40
Inverse flattening
F
−1 = A 1 /(A 1 − A 2 )
298.257 222 101
298.256 420 489