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surfaces include (i) the plane, (ii) the sphere, (iv) the ellipsoid-of-revolution, and (iv) the triaxial
ellipsoid.
Section J-2.
More specific, Sect. 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
Grobner 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 twenty-one points varying from Finland, Sweden, Lithuania, Poland, and Germany and taking
reference to the World Geodetic Datum 2000 with the
data A 1 =6 , 378 , 136 . 602 m (semi-
major axis) and A 2 =6 , 656 , 751 . 860 m (semi-minor axis) following Grafarend and Ardalan ( 1999 ).
{
A 1 ,A 2 }
Section J-3.
Finally, Sect. 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.
J-1 Projective Heights in Geometry Space: From Planar/Spherical
to Ellipsoidal Mapping
3 kl }
are connected by geodesics , namely by straight lines
which are derived from a variational principle . The general solution of the differential equations
of a geodesic, in particular, in terms of an a ne parameter of its length, is represented by a linear
one-dimensional manifold embedded into { R
First, we outline how points in
{ R
3 kl } .
in Fig. J.1 , we introduce the orthogonal projection of a point P (the peak of a mou nta in, the top
of a tower) onto a horizontal plane through P 0 by p = π ( P ) generating the height pP of P with
respect to the plane P
3 kl } . Second, based upon geodesics in { R
0 through P 0 . Alternatively, we may interpret the orthogonal projection
p = π ( P ) along a geodesic/straight line as a minimal distance mapping of P with respect to
P 0 generating p = π ( P ). Third, by Fig. J.2 , we illustrate the minimal distance mapping of a
topographic point P
2 as an element of the topographic surface (two-dimensional Riemann
manifold) of the Earth along a geodesic/straight line onto a plane
T
2 which may be chosen as
the horizontal plane at some reference point. In this wa y, we generate the orthogonal projection
p = π ( P ) and the length of the short est distance pP , called the geometric height of P with
respect to
P
2 . The choice of the height pP is very popular in photogrammetric and engineering
surveying. By contrast, by Figs. J.3 and J.4 , we illustrate the minimal distance mapping of a
point P onto p
P
2
2
A 1 ,A 2
2
A 1 ,A 2 ,A 3
along a geodesic/straight line through P .Let
us here assume that the reference surface is no longer the plane
S
r or p
E
or p
E
2 , but the sphere
2
P
S
r of radius r
2
A 1 ,A 2
or the ellipsoid-of-revolution
E
of semi-major axis A 1 and semi-minor axis A 2 or the triaxial
 
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