Geodetic Mapping - Map Projections: Cartographic Information Systems - page 501

I AGK ( B N = −B S )= 1
20

l E

E 2 ) 2 × (1 −

1

3 sin 2 B N (1 + E 2 )+sin 4 B N 4 15 E 2 sin 6 B N +O 2 ( E 4 ) +

2

9 E 2 + 1

2

× − − (20.148)

5

+O GK ( l E ) .
Parallel coordinates of type Soldner:
I AS = 1

2 I AGK +O S ( l E ) .

(20.149)

Normal coordinates of type Riemann:

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 −

b N − b S b N − b S

+ D

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

(20.150)
× 1 − 2 E 2 sin 2 B 0 +sin B 0 (1 − 2 E 2 sin 2 B 0 − 4 E 2 cos 2 B 0 )( b N + b S ) / 2cos B 0 +
+O R (6) ,

×

12 b N − 6 b N − b S

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

b S )= cos 4 B 0
360

I AR ( b N = −

1+O( E 4 ) b N l E + (1

+ cos 2 B 0
324

2 E 2 cos 2 B 0 +O( E 4 ))
360

−

b N + (20.151)

+O R (6) .
(A,B,C,D,P,Qaregivenby( 20.173 )).

End of Corollary.

Proof (Gauss-Krueger, Λ 1 = Λ 2 = Λ, l := L − L 0 ).

d S E 2 A 1 ,A 2

+ l E B N

1) 2 = A 1 (1 − E 2 )
S E 2 A 1 ,A 2

cos B ( Λ − 1) 2
(1

1
S E 2 A 1 ,A 2

I AGK = ( Λ − d l d B

E 2 sin 2 B 2 ) 2 ,

(20.152)

−

−l E B S

E 2 sin 2 B
1

1= 1

2 cos 2 B 1

−

l 2 + Λ −

− E 2

+O GK ( l 4 ) ,

(20.153)

E 2 sin 2 B ) 2
(1

( Λ − 1) 2 = 1

4 cos 4 B (1

−

l 4 +

− E 2 ) 2

+O GK ( l 6 ) ,

⎡ ⎤

d S E 2 A 1 ,A 2 ( Λ − 1) 2 = 1

+ l E B N

A 1

⎣ l 4 ⎦ =

d B cos 5 B +O GK ( l 6 ) d l

4 1 − E 2

−l E B S