Geography Reference
In-Depth Information
Λ 1 2 }
of type ( 23.8d )and( 23.8f ), respectively, are computed by means of solving the characteristic
equations ( 23.8c )and( 23.8e ), respectively. The fourth step leads us to the postulates of
(
by ( 23.7b )and( 23.7c ), respectively, and C l given by ( 23.8b ). As a result the eigenvalues
{
α
) a conformal mapping (conformeomorphism): Box 23.3 Λ 1 = Λ 2 ,
(
β
) an equiareal mapping (areomorphism): Box 23.4 Λ 1 Λ 2 =1,
) an equidistant mapping: Box 23.5 Λ 2 =1 .
For the case ( α )ofa conformal mapping the “ canonical postulate Λ 1 = Λ 2 ,theidentityof
the eigenvalues (left principal stretches) leads us to ( 23.9a )and( 23.9b )as first order differential
equations, which are solved by ( 23.9c ), ( 23.9d )aswellas( 23.9e ), respectively. The integration
constants c f and c g , respectively, are fixed by boundary conditions ( 23.9f )and( 23.9h ), respectively,
such that the final mapping equations ( 23.9g )and( 23.9i )appear.
(
γ
Box 23.2 (The generic steps of a map projection for a given structure of the map ( 23.4 )of
type plane parallel to T π/ 2 M
2 and ( 23.5 ) of type circular cylinder C
A + B ).
The left Jocabi map
(23.6a)
J l := x U x V
=
f sin Uf cos U
f cos Uf sin U
(23.6b)
y U y U
J l := x U x V
= A + B 0
(23.6c)
g
y U y U
0
Subject to
the orientation conservation
f < 0
g > 0
J l = x U y V
x V y U > 0
(23.6d)
TheleftCauchy - Green map
(23.7a)
C l := J l G r J l = f 2
0 f 2
0
(23.7b)
C l := J l G r J l = ( A + B ) 2 0
g 2
(23.7c)
0
Subject to
the right metric G r = I 2 of the tangent plane
and
the developed circular cylinder
The general eigenvalue problem
(23.8a)
Λ 2 G l =0 subject to ( 23.7a ) G l = ( A + B cos V ) 2
B 2
C l
0
(23.8b)
0
Λ 2 G l =
Λ 2 B 2 2
C l
− Λ 2 ( A + B cos V ) 2
f 2
0
=0
(23.8c)
f 2
0
 
Search WWH ::




Custom Search