Gauss Surface Normal Coordinates in Geometry and Gravity - Map Projections: Cartographic Information Systems - page 839

twenty-one points of the topographic surface of the Earth presented in Table J.1 . Compare with
Fig. J.6 .

Tab l e J . 1 Cartesian coordinates of topographic point (Baltic Sea Level Project)

Station X (m) Y (m) Z (m)

Borkum (Ger) 3,770,667.9989 446,076.4896 5,107,686.2085

Degerby (Fin) 2,994,064.9360 1,112,559.0570 5,502,241.3760

Furuogrund (Swe) 2,527,022.8721 981,957.2890 5,753,940.9920

Hamina (Fin) 2,795,471.2067 1,435,427.7930 5,531,682.2031

Hanko (Fin) 2,959,210.9709 1,254,679.1202 5,490,594.4410

Helgoland (Gcr) 3,706,044.9443 513,713.2151 5,148,193.4472

Helsinki (Fin) 2,885,137.3909 1,342,710.2301 5,509,039.1190

Kemi (Fin) 2,397,071.5771 1,093,330.3129 5,789,108.4470

Klagshamn (Swe) 3,527,585.7675 807,513.8946 5,234,549.7020

Klaipeda (Lit) 3,353,590.2428 1,302,063.0141 5,249,159.4123

List/Sylt (Gcr) 3,625,339.9221 537,853.8704 5,202,539.0255

Molas (Lit) 3,358,793.3811 1,294,907.4149 5,247,584.4010

Mantyluoto (Fin) 2,831,096.7193 1,113,102.7637 5,587,165.0458

Raahe (Fin) 2,494,035.0244 1,131,370.9936 5,740,955.4096

Ratan (Swe) 2,620,087.6160 1,000,008.2649 5,709,322.5771

Spikarna (Swe) 2,828,573.4638 893,623.7288 5,627,447.0693

Stockholm (Swe) 3,101,008.8620 1,013,021.0372 5,462,373.3830

Ustka (Pol) 3,545,014.3300 1,073,939.7720 5,174,949.9470

Vaasa (Fin) 2,691,307.2541 1,063,691.5238 5,664,806.3799

Visby(Swc) 3,249,304.4375 1,073,624.8912 5,364,363.0732

Olands N. U. (Swe)

3,295,551.5710 1,012,564.9063 5,348,113.6687

From the algorithm of Box J.8 , the first polynomial equation of fourth order of the Grobner
basis is obtained as ( J.72 ). Numerical values are provided by Table J.2 .

c 1 x 4 + c 3 x 4 + c 2 x 4 + c 1 x 4 + c 0 =0 ,
c 4 = a 6 b 6 ,c 3 =2 a 6 b 4 +2 a 4 b 6 ,
c 2 = a 6 b 2 +4 a 4 b 4 + a 2 b 6

− a 4 b 2 X 2 − a 4 b 2 Y 2 − a 2 b 4 Z 2 ,

(J.72)

c 1 =2 a 4 b 2 +2 a 2 b 4 2 a 2 b 2 X 2 2 a 2 b 2 Y 2 2 a 2 b 2 Z 2 , − − −

c 0 = a 2 b 2 b 2 X 2 b 2 Y 2 a 2 Z 2 .

− − −

J-3 Gauss Surface Normal Coordinates: Case Study Triaxial Ellipsoid

First, we review surface normal coordinates for the triaxial ellipsoid. Second, it is our duty to
review representative data for the triaxial ellipsoid for the Earth and other celestial bodies.