640 J Gauss surface normal coordinates in geometry and gravity space
X
x
Fig. J.3. Projective heights in geometry space, minimal distance mapping with respect to a reference sphere
S
2
r , spherical heights h S (length of the geodesic from X ∈ T
2 to x ∈ S
2
r ).
Let us calculate the projection heights in geometry space, in detail, the minimal distance with
respect to a reference surface of type (i) plane P
2 , (ii) sphere S
2
r , (iii) ellipsoid-of-revolution E
2
A 1 ,A 2
,
and (iv) triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
in Box J.2.
X
x
Fig. J.4. Projective heights in geometry space, minimal distance mapping with respect to a reference triaxial
ellipsoid E
2
A 1 ,A 2 ,A 3 or a reference ellipsoid-of-revolution E
2
A 1 ,A 2 , ellipsoidal heights h E (length of the geodesic
from X ∈ T
2 to x ∈ E
2
A 1 ,A 2 or E
2
A 1 ,A 2 ,A 3 ).