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with the “ left Cauchy-Green deformation tensor C l , namely by a simultaneous diagonalization of
the pair of positive definite matrices
, we have established the third generic step as the
general eigenvalue problem ( 23.64 ) with respect to the matrices C l given by ( 23.62 )and( 23.63 ),
respectively, and G l given by ( 23.65 ). The eigenvalues
{
C l ,G l }
of type ( 23.67 )and( 23.69 ),
respectively, are computed by means of solving the characteristic equations ( 23.66 )and( 23.68 ),
respectively. The fourth step leads us to the postulate of
( α ) a conformal mapping (conformeomorphism): Box 23.18 Λ 1 = Λ 2 ,
( β ) an equiareal mapping (areomorphism): Box 23.19 Λ 1 Λ 2 =1,
( γ ) an equidistant mapping: Box 23.20 Λ 2 =1.
{
Λ 1 , Λ 2 }
Box 23.17 (The generic steps of a map projection for a given structure of the map ( 23.55 )
and ( 23.56 ) of type circular cylinder and circular cone
2
C
0 ).
The left Jacobi map
(23.57)
J l i := x U x V
= 10
(23.58)
0 f
y U y V
J l ii := α U α V
= n 0
(23.59)
0 g
r U r V
Subject to the orientation conservation
|
f s
J l |
= x U y V
x V y U > 0
(23.60)
g > 0
|
J l |
= α U r V
α V r U > 0
TheleftCauchy - Green map
(23.61)
C l i := J l i G r i J l i = 10
0 f 2
G r i = I 2
(23.62)
C l ii := J l ii G r ii J l ii = n 2 g 2 0
0 g 2
∀G r ii = r 2 0
(23.63)
01
subject to the right metric G r i = I 2 of the developed cone
and
the right metric G r ii = Diag ( r 2 , 1) of the developed cone
The general eigenvalue problem
(23.64)
Λ 2 G l =0 subject to ( 23.61 ) G l = V 2
01+4 V 2
C l
0
(23.65)
2 G l = 1
=0
C l i
2 V 2
0
(23.66)
f 2
2 (1 + 4 V 2 )
0
f
1+4 V 2
Λ 1 = 1
V , Λ 2 =
(23.67)
Λ 2 G l = n 2 g 2
=0
C l ii
− Λ 2 V 2
0
(23.68)
0
g 2
− Λ 2 (1 + 4 V 2 )
g
Λ 1 = ng
1+4 V 2
V , Λ 2 =
(23.69)
 
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