J Gauss surface normal coordinates in geometry and gravity
space Three-dimensional geodesy, minimal distance mapping, geometric heights. Reference plane, reference
sphere reference ellipsoid-of-revolution, reference triaxial ellipsoid.
With the advent of artifical satellites in an Earth-bound orbit, geodesists succeeded to position points
of the topographic surface T
2 by a set of {X, Y, Z} ∈ R
3 coordinates in a three-dimensional reference
frame at the mass center of the Earth oriented along the equatorial axes at some reference epoch t 0 ∈ R.
In particular, global positioning systems (“global problem solver”: GPS), were responsible for the
materialization of three-dimensional geodesy in an Euclidean space. Based upon a triple {X, Y, Z} ∈ R
3
of coordinates new concepts for converting these coordinates into heights with respect to a reference
surface have been developed.
Important!
In the geometry space, the triplet {X, Y, Z} ∈ T
2 is transformed by a geodesic projection
into geometric heights with respect to (i) a reference plane P
2 , (ii) a reference sphere S
2
r , (iii)
a reference ellipsoid-of-revolution E
2
A 1 ,A 2
, or (iv) a reference triaxial ellipsoid E
2
A 1 .A 2 ,A 3
.
Important!
First, the geodesic projection is performed by a straight line as the geodesic in flat geometry space. Second, the special geodesic passing the point {X, Y, Z} ∈ T has been chosen
which has minimal distance S to the reference surface. The length of the geodesic from
{X, Y, Z} ∈ T to {ˆ x, ˆ
y, ˆ
z} ∈ P
2 or S
2
r or E
2
A 1 ,A 2
or E
2
A 1 .A 2 ,A 3
, in short, {ˆ x, ˆ
y, ˆ
z} being determined by the minimal distance mapping, constitute the projective height in geometry space,
namely of type (i) planar, (ii) spherical, (iii) ellipsoidal, or (iv) triaxial ellipsoidal.
Section J-1.
By algebraic mean, Section J-1 outlines various step procedures to establish projection heights in
geometry space. By means of minimal distance mapping, various computational steps, either forward
or backwards, are reviewed depending on the nature of the projection surface. The projection surfaces
include (i) the plane, (ii) the sphere, (iv) the ellipsoid-of-revolution, and (iv) the triaxial ellipsoid.
Section J-2.
More specific, Section J-2 reviews various algorithms of computing Gauss surface normal coordinates
for the case of an ellipsoid-of-revolution. The highlight is the computational algorithm by means of
Gr¨ obner basis and the Buchberger algorithm in establishing an ideal for the polynomial solution for the
minimum distance mapping. From the Baltic Sea Level Project, we refer to detailed solutions of twentyone points varying from Finland, Sweden, Lithuania, Poland, and Germany and taking reference to
the World Geodetic Datum 2000 with the {A 1 , A 2 } data A 1 = 6 378 136.602 m (semi-major axis) and
A 2 = 6 656 751.860 m (semi-minor axis) following E. Grafarend and A. Ardalan (1999).
Section J-3.
Finally, Section J-3 presents the computation of Gauss surface normal coordinates for the case of a
triaxial ellipsoid. For the Earth, we compute the position and orientation, and from parameters of the
best fitting triaxial ellipsoid, we chose the geoid as the ideal Earth figure closest to the mean sea level.
This important result is extended to other celestial bodies of triaxial nature, namely for Moon, Mars,
Phobos, Amalthea, Io, and Mimas.
space Three-dimensional geodesy, minimal distance mapping, geometric heights. Reference plane, reference
sphere reference ellipsoid-of-revolution, reference triaxial ellipsoid.
With the advent of artifical satellites in an Earth-bound orbit, geodesists succeeded to position points
of the topographic surface T
2 by a set of {X, Y, Z} ∈ R
3 coordinates in a three-dimensional reference
frame at the mass center of the Earth oriented along the equatorial axes at some reference epoch t 0 ∈ R.
In particular, global positioning systems (“global problem solver”: GPS), were responsible for the
materialization of three-dimensional geodesy in an Euclidean space. Based upon a triple {X, Y, Z} ∈ R
3
of coordinates new concepts for converting these coordinates into heights with respect to a reference
surface have been developed.
Important!
In the geometry space, the triplet {X, Y, Z} ∈ T
2 is transformed by a geodesic projection
into geometric heights with respect to (i) a reference plane P
2 , (ii) a reference sphere S
2
r , (iii)
a reference ellipsoid-of-revolution E
2
A 1 ,A 2
, or (iv) a reference triaxial ellipsoid E
2
A 1 .A 2 ,A 3
.
Important!
First, the geodesic projection is performed by a straight line as the geodesic in flat geometry space. Second, the special geodesic passing the point {X, Y, Z} ∈ T has been chosen
which has minimal distance S to the reference surface. The length of the geodesic from
{X, Y, Z} ∈ T to {ˆ x, ˆ
y, ˆ
z} ∈ P
2 or S
2
r or E
2
A 1 ,A 2
or E
2
A 1 .A 2 ,A 3
, in short, {ˆ x, ˆ
y, ˆ
z} being determined by the minimal distance mapping, constitute the projective height in geometry space,
namely of type (i) planar, (ii) spherical, (iii) ellipsoidal, or (iv) triaxial ellipsoidal.
Section J-1.
By algebraic mean, Section J-1 outlines various step procedures to establish projection heights in
geometry space. By means of minimal distance mapping, various computational steps, either forward
or backwards, are reviewed depending on the nature of the projection surface. The projection surfaces
include (i) the plane, (ii) the sphere, (iv) the ellipsoid-of-revolution, and (iv) the triaxial ellipsoid.
Section J-2.
More specific, Section J-2 reviews various algorithms of computing Gauss surface normal coordinates
for the case of an ellipsoid-of-revolution. The highlight is the computational algorithm by means of
Gr¨ obner basis and the Buchberger algorithm in establishing an ideal for the polynomial solution for the
minimum distance mapping. From the Baltic Sea Level Project, we refer to detailed solutions of twentyone points varying from Finland, Sweden, Lithuania, Poland, and Germany and taking reference to
the World Geodetic Datum 2000 with the {A 1 , A 2 } data A 1 = 6 378 136.602 m (semi-major axis) and
A 2 = 6 656 751.860 m (semi-minor axis) following E. Grafarend and A. Ardalan (1999).
Section J-3.
Finally, Section J-3 presents the computation of Gauss surface normal coordinates for the case of a
triaxial ellipsoid. For the Earth, we compute the position and orientation, and from parameters of the
best fitting triaxial ellipsoid, we chose the geoid as the ideal Earth figure closest to the mean sea level.
This important result is extended to other celestial bodies of triaxial nature, namely for Moon, Mars,
Phobos, Amalthea, Io, and Mimas.