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C l and G l are the left Cauchy-Green matrix and the left metric matrix, respectively.
End of Lemma.
Proof.
det[C l − Λ S G l ]=0
(18.4)
Λ S − Λ S tr [C l G l ]+det[C l G l ]=0 ,
( Λ S ) + ( Λ S ) =1
(canonical postulate of equal area mapping)
det [C l ] / det [G l ] = 1
(18.5)
(Lemma of Vieta:
the product of the solutions of a quadratic equation equals the absolute term) .
End of Proof.
From the general form of the deformation tensor , the metric tensor, and the postulate of equal
area mapping, we derive the general structure of equal area pseudo-conic mappings of the sphere
in Box 18.1 . As a side remark, we use the result that only pseudo-conic projections of type equal
area exist .
Box 18.1 (General structure of equal area pseudo-conic mappings of the sphere).
Mapping equations:
α = g ( Δ ) Λ cos Δ = h ( Δ ) Λ, r = f ( Δ ) .
(18.6)
Left Cauchy-Green matrix:
C l =J l G r J l = f 2 h 2 f 2 hh Λ
,
(18.7)
f 2 hh Λf 2 h 2 Λ 2 + f 2
det[C l ]= f 2 h 2 ( f 2 h 2 Λ 2 + f 2 )
f 4 h 2 h 2 Λ 2 = f 4 h 2 h 2 Λ 2 + f 2 h 2 f 2
f 4 h 2 h 2 Λ 2 = f 2 f 2 h 2 .
(18.8)
Left Jacobi matrix:
J l = D Λ αD Δ α
= h ( Δ ) Λh ( Δ )
.
(18.9)
f ( Δ )
D Λ rD Δ r
0
 
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