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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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