Geography Reference
In-Depth Information
We gain the parametric representation by Λ = Λ and X 2 + Y 2 = X 2 + Y 2 . The integral
dependent of the reduced latitude then is obtained as follows.
:= ( X,Y,Z ): X 2 + Y 2
A 2 =1
+ Z 2
2
A 1 ,A 2
3 ,
E
R
(14.10)
A 1
A 1 = 1
Z
X 2 + Y 2 =(1
E 2 )tan Φ = A 2
A 2
A 1 tan Φ ,
E 2
tan Φ = 1 − E 2 tan Φ = A 2
A 1 tan Φ
(14.11)
E 2 ) Φ
0
d Φ
f ( Φ )= A 1 (1
E 2 sin 2 Φ ) 3 / 2
(1
= A 1 Φ
0
d Φ (1
E 2 cos 2 Φ ) .
(14.12)
Let us here also define the elliptic integral of the second kind (for example, consult Appendix C
or Gradsteyn and Ryzhik 1983 , namely p. 905, formula 8.1113). The definition ( 14.13 )leadsfor
f ( Φ ) to the representation ( 14.14 ). The principal stretches are easily computed as ( 14.15 ).
E 2 ) Φ
0
E 2 sin 2 Φ ) 3 / 2 , f ( Φ )= A 1 Φ
d Φ 1
d Φ
f ( Φ )= A 1 (1
E 2 cos 2 Φ ,
(1
0
f ( Δ )= A 1 π/ 2
Δ
E 2 sin 2 Δ = A 1 π/ 2
0
d Δ 1
d Δ 1
E 2 sin 2 Δ
(14.13)
A 1 Δ
0
d Δ 1
E 2 sin 2 Δ = A 1 [ E ( π/ 2 ,E )
E ( Δ ,E )] ,
f ( Φ )= A 1 E ( π/ 2 ,E ) E π
2 arctan A 2
A 1 tan Φ ,E ,
(14.14)
Λ 1 = 1 E 2 sin 2 Φ
cos Φ
, Λ 2 =1 .
(14.15)
14-22 Special Normal Cylindric Mapping (Normal Conformal,
Equidistant: Equator)
First, let us here apply the postulate of conformal mapping. Similar as before, we obtain the
following set of formulae.
 
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