J-2 Gauss surface normal coordinates: case study ellipsoid-of-revolution 651
Let us adopt the World Geodetic Datum 2000 with the data “semi-major” axis A 1 = 6 378 136.602 m
and “semi-major axis” A 2 = 6 356 751.860 m of the International Reference Ellipsoid (E. Grafarend
and A. Ardalan 1999). Here, we take advantage of given Cartesian coordinates of twenty-one points
of the topographic surface of the Earth presented in Table J.1. Compare with Fig. J.6.
Table J.1. Cartesian coordinates of topographic point (Baltic Sea Level Project).
station
X [m]
Y [m]
Z [m]
Borkum (Ger)
3770667.9989
446076.4896
5107686.2085
Degerby (Fin)
2994064.9360
1112559.0570
5502241.3760
Furu¨ ogrund (Swe)
2527022.8721
981957.2890
5753940.9920
Hamina (Fin)
2795471.2067
1435427.7930
5531682.2031
Hanko (Fin)
2959210.9709
1254679.1202
5490594.4410
Helgoland (Gcr)
3706044.9443
513713.2151
5148193.4472
Helsinki (Fin)
2885137.3909
1342710.2301
5509039.1190
Kemi (Fin)
2397071.5771
1093330.3129
5789108.4470
Klagshamn (Swe)
3527585.7675
807513.8946
5234549.7020
Klaipeda (Lit)
3353590.2428
1302063.0141
5249159.4123
List/Sylt (Gcr)
3625339.9221
537853.8704
5202539.0255
Molas (Lit)
3358793.3811
1294907.4149
5247584.4010
M¨ antyluoto (Fin)
2831096.7193
1113102.7637
5587165.0458
Raahe (Fin)
2494035.0244
1131370.9936
5740955.4096
Ratan (Swe)
2620087.6160
1000008.2649
5709322.5771
Spikarna (Swe)
2828573.4638
893623.7288
5627447.0693
Stockholm (Swe)
3101008.8620
1013021.0372
5462373.3830
Ustka (Pol)
3545014.3300
1073939.7720
5174949.9470
Vaasa (Fin)
2691307.2541
1063691.5238
5664806.3799
Visby(Swc)
3249304.4375
1073624.8912
5364363.0732
¨
Olands N. U. (Swe)
3295551.5710
1012564.9063
5348113.6687
From the algorithm of Box J.8, the first polynomial equation of fourth order of the Gr¨ obner basis
is obtained as (J.72). Numerical values are provided by Table J.2.
Table J.2. Polynomial coefficients of the univariate polynomial of order four in x 4 .
point
c 0
c 1
c 2
c 3
c 4
1
−2.3309099e + 22
1.334253e + 41
1.351627e + 55
4.382358e + 68
4.441958e + 81
2
−1.142213e + 22
1.3351890e + 41
1.352005e + 55
4.382358e + 68
4.441958e + 81
3
−1.720998e + 22
1.335813e + 41
1.352259e + 55
4.382358e + 68
4.441958e + 81
4
−8.871288e + 21
1.335264e + 41
1.352035e + 55
4.382358e + 68
4.441958e + 81
5
−1.308070e + 22
1.335160e + 41
1.351993e + 55
4.382358e + 68
4.441958e + 81
6
−2.275210e + 22
1.334345e + 41
1.351665e + 55
4.382358e + 68
4.441958e + 81
7
−1.272935e + 22
1.335205e + 41
1.352012e + 55
4.382358e + 68
4.441958e + 81
8
−1.373946e + 22
1.335906e + 41
1.352296e + 55
4.382358e + 68
4.441958e + 81
9
−1.981047e + 22
1.334546e + 41
1.351746e + 55
4.382358e + 68
4.441958e + 81
10
−2.755981e + 22
1.334574e + 41
1.351758e + 55
4.382358e + 68
4.441958e + 81
11
−2.330047e + 22
1.334469e + 41
1.351715e + 55
4.382358e + 68
4.441958e + 81
12
−1.538357e + 22
1.334580e + 41
1.351759e + 55
4.382358e + 68
4.441958e + 81
13
−1.117760e + 22
1.335399e + 41
1.352090e + 55
4.382358e + 68
4.441958e + 81
14
−1.124559e + 22
1.335785e + 41
1.352246e + 55
4.382358e + 68
4.441958e + 81
15
−1.200556e + 22
1.335704e + 41
1.352214e + 55
4.382358e + 68
4.441958e + 81
16
−1.427443e + 22
1.335496e + 41
1.352130e + 55
4.382358e + 68
4.441958e + 81
17
−1.836471e + 22
1.335087e + 41
1.351965e + 55
4.382358e + 68
4.441958e + 81
18
−1.772332e + 22
1.334410e + 41
1.351690e + 55
4.382358e + 68
4.441958e + 81
19
−1.012020e + 22
1.335593e + 41
1.352168e + 55
4.382358e + 68
4.441958e + 81
20
−1.427711e + 22
1.334856e + 41
1.351870e + 55
4.382358e + 68
4.441958e + 81
21
−1.644250e + 22
1.334815e + 41
1.351854e + 55
4.382358e + 68
4.441958e + 81