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being positive-definite leads
to the general eigenvalue problem ( F.15 ), leading to the left principal stretches ( F.16 ), solved
by ( F.17 ), subject to ( F.18 ).
Simultaneous diagonalization of the two matrices
{
c AB }
and
{
G AB }
=0=
Λ S M 2
Λ S L 2 ΛLg ( L g + Lg )
ΛLg ( L g + Lg ) Λ 2 ( L g + Lg ) 2 + f 2
L 2 g 2
Λ S G AB |
|
c AB
,
(F.15)
Λ S Λ 2 ( L g + Lg ) 2
M 2 + g 2 + g 2 f 2
+ f 2
Λ S
=0 ,
(F.16)
M 2
M 2
Λ 1 = 1
4 b ) , Λ 2 = 1
2 ( a + a 2
a 2
2 ( a
4 b ) ,
(F.17)
a := Λ 2 ( L g + Lg ) 2 /M 2 + f 2 /M 2 + g 2 ,
b := g 2 f 2 /M 2 .
(F.18)
The postulate of equiareal mapping Λ 1 Λ 2 =1leadsto( F.19 ) being equivalent to b =1or
g = Mf 1 or g 2 f 2 /M 2 = 1. Obviously, the postulate of a conformal mapping Λ 1 = Λ 2 cannot
be fulfilled since a 2 =4 b leads to a nonlinear functional F ( g 4 ,g 2 ,g 4 ,g 2 ,f 4 ,f 2 ).
2 ( a + a 2
2 ( a − a 2
1
4 b ) 1
4 b ) = 1
(F.19)
End of Proof.
In summarizing, we are led to the equiareal pseudo-cylindrical mapping equations of Box F.2 .
Box F.2 (Equiareal pseudo-cylindrical mapping).
x = A 2 (1
E 2 )cos Φ
1
f ( Φ ) = M ( Φ ) N ( Φ )cos ΦΛ
1
f ( Φ ) ,
E 2 sin 2 Φ ) 2 Λ
(1
(F.20)
y = f ( Φ ) .
F-2 Mixed Equiareal Cylindric Mapping: Biaxial Ellipsoid Onto Plane
The variants of mixed equiareal mappings of the biaxial ellipsoid onto the plane are generated
by the weighted mean of the normal equiareal cylindric mapping (for the sphere S
2 R this is the
Lambert equiareal projection) and the equiareal pseudo-cylindrical mapping of sinusoidal type
(for the sphere S
2 R this is the Sanson-Flamsteed equiareal projection). The separate mappings are
beforehand collected in Corollary F.2 .
Corollary F.2 (Ellipsoidal equiareal cylindric projection of normal type).
The ellipsoidal equiareal cylindric projection of normal type (generalized Lambert projection) in
the class of cylindric projections in terms of surface normal longitude Λ and latitude Φ of E
2 A,B is
represented by ( F.21 ) mapping the circular equator equidistantly.
 
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