Geodetic Mapping - Map Projections: Cartographic Information Systems - page 505

P := − sin B 0 cos 2 B 0
9

[1 + 2 E 2 (1 − 2cos 2 B 0 )] + O( E 4 ) ,

Q := −

sin B 0
36

[1 + 2 E 2 (1 5cos 2 B 0 )] + O( E 4 ) ,

−

A

I AR = A 1 (1 − E 2 )
S E 2 A 1 A 2 5 l F ( b N − b S )+ B 9 l E ( b N − b S )+

(20.174)

b S )+O(8) , + C b S )+ D b S )+ P b S )+ Q

5 l E ( b N − 10 l E ( b N − 12 l E ( b N − 6 l E ( b N −

1
S E 2 A 1 ,A 2

=

2 E 2 sin 2 B 0 ++sin B 0 (1 2 E 2 sin 2 B 0 −

= 1

− − 4 E 2 cos 2 B 0 )( b N + b S ) / 2cos B 0 +O( b 2 )

,

2 A 1 (1

− E 2 )cos B 0 l E ( b N − b S )

(20.175)

A

b N − b S b N − b S

1
2cos B 0 5 l E + B b S l E + C

I AR =

b S +

9 b N − 5 b N −

12 P b N −

b S b N − b S

+ D

10 ( b N + b S ) l E + P b S l E + Q

× (20.176)

b N − 6 b N − b S

2 E 2 sin 2 B 0 +sin B 0 (1 2 E 2 sin 2 B 0 −

1 − − 4 E 2 cos 2 B 0 )( b N + b S )

× +O R (6) ,

2cos B 0

b S )= cos 4 B 0
360

(1 + 2 E 2 cos 2 B 0 +O( E 4 )) l E +

I AR ( b N = −

+ cos 2 B 0
324

(1 + O( E 4 )) b N l E + (1 − 2 E 2 cos 2 B 0 +O( E 4 ))

b N +O R (6) . (20.177)

360

End of Proof.

As an example, we compare in Table 20.5 the total deformation energy for conformal Gauss-
Krueger coordinates, parallel Soldner coordinates, and normal Riemann coordinates. Obviously,
in the range of application, normal Riemann coordinates generate the smallest global distortion ,
followed by parallel Soldner coordinates (factor half compared to conformal Gauss-Krueger coor-
dinates), and conformal Gauss-Krueger coordinates.
Here, we take reference to G. B. Airy's definition of “balance of erros”. Conformal geodesics were
treated by Beltrami ( 1869 ), Fialkow ( 1939 ), Legendre ( 1806 ), Schouten ( 1954 ), Vogel ( 1970 , 1973 ),
Weingarten ( 1861 ), and Yano ( 1970 , 1940a , b ). Boltz ( 1943 ) presented formulae and tables for the
normal computation of Gauss-Krueger coordinates which we use here. For the optimal Universal