652 J Gauss surface normal coordinates in geometry and gravity space
A 1
A 2
p
P
spray at P , orthogonal projection p
Fig. J.6. Minimum distance mapping of a point P on the Earth’s topographic surface to a point p on the
International Reference Ellipsoid E
2
A 1 ,A 2 .
c 4 x
4
4 + c 3 x
3
4 + c 2 x
2
4 + c 1 x 4 + c 0 = 0 ,
c 4 = a
6 b
6 , c 3 = 2a
6 b
4 + 2a
4 b
6 ,
c 2 = a
6 b
2 + 4a
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 ,
c 1 = 2a
4 b
2 + 2a
2 b
4
− 2a
2 b
2 X
2
− 2a
2 b
2 Y
2
− 2a
2 b
2 Z
2 ,
c 0 = a
2 b
2
− b
2 X
2
− b
2 Y
2
− a
2 Z
2 .
(J.72)
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.
J-31 Review of surface normal coordinates for the triaxial ellipsoid
In case of a triaxial ellipsoid, we depart from the representation (J.73) subject to (J.74) once we use
surface normal coordinates.
X
2
A 2
1
+
Y
2
A 2
2
+
Z
2
A 2
3
= 1 ,
⎡
⎣
X
2
Y
2
Z
2
⎤
⎦ =
A
2
1
W
⎡
⎣
cos B cos L
1 − E
2
12
cos B cos L
1 − E
2
13
sin B
⎤
⎦ ,
(J.73)
W = W (L, B) :=
1 − E 2
13 sin
2 B − E 2
12 cos 2 B sin
2 L .
(J.74)