Geodetic Mapping - Map Projections: Cartographic Information Systems - page 487

The construction principle is as follows. A point P 0 ( L 0 ,B 0 ) is defined as the origin of a local
reference system .The x c axis of the coordinate system, which is called “Hochwert” (Soldner
wording), agrees with the chosen reference meridian L 0 . A point P ( L, B ) is described by geodetic
parallel coordinates as follows. Through the point P , we compute the unique geodetic line ,which
cuts the meridian L 0 at a right angle to produce the footprint point P F . The length of the geodesic
P - P F is chosen as the y c coordinate, which is called “Rechtswert” (Soldner wording). The angle
which is in between the local meridian of the point P and the geodetic parallel through the point
P is called meridian convergence γ = A PF −π/ 2. The angle γ is fixed as the northern part of the
meridian, lefthand-oriented positive. We always say that the azimuth of the coordinate line y c =
const. is the angle γ . Most notable, the geodetic parallel y c =const.is not a geodesic. Compare
with Fig. 20.4 .

The y c lines produce the geodesic field . In contrast, the
geodetic parallels are the orthogonal trajectories d s 2

=

E ( x c ,y c )d x c +d y c =d x 2 + G ( x, y )d y 2 .

20-31 First Problem of Soldner Coordinates: Geodetic Parallel
Coordinates, Input { L 0 , B 0 , x c = y, y c = x } Versus Output

{ L , B ,γ (Meridian Convergence) }

For the problem of given coordinates { L 0 ,B 0 ,x c = y, y c = x } and unknown coordinates

, we use the standard method of Legendre series u = s cos α 0 P and v = s sin α 0 P .
In Box 20.7 , the details are collected.

{ L, B, γ }

Box 20.7 (The problem of given coordinates {L 0 ,B 0 ,x c = y, y c = x} and unknown
coordinates {L, B, γ} ).
First, we rewrite ( 20.73 )-( 20.75 )intermsof u = s cos α 0 P and v = s sin α 0 P and obtain
(the coecients [ μν ] are collected in Box 20.5 and have to be computed at the point B 0 )

B P = B 0 + [01] v versus L P = L 0 + [10] u versus α P 0 = α 0 P + [10] α u

+[20] u 2

+[11] uv +[11] α uv

+[02] v 2 +[12] uv 2 +[12] α uv 2

+[21] u 2 v +[30] u 3 +[30] α u 3

+[03] v 3 +[31] u 3 v +[31] α u 3 v

(20.112)

+[40] u 4 +[13] uv 3 +[13] α uv 3

+[22] u 2 v 2

+ ... + ....

+[04] v 4
+ ...

First step: determine L F ,B F ,α F0 , starting point P 0 ,s = y = x c ,x = y c =0 ,α F0 = π/ 2
given.