580 F Mixed cylindric map projections
Proof (of (F.47)).
For the proof of (F.47), we depart from the general eigenvalue problem (F.15), whose characteristic
equation is solved under the postulate of equal area Λ 1 Λ 2 = 1, which is equivalent to det
C l G
−1
l
= 1.
c AB − Λ
2
S G AB
= 0
⇒
Λ
4
S − Λ
2
S tr
C l G
−1
l
+ det
C l G
−1
l
= 0 ,
(F.50)
Λ
2
1,2 =
1
2
tr
C l G
−1
l
±
tr
C l G
−1
l
2 − 4
det
C l G
−1
l
⇒
det
C l G
−1
l
= 1 .
(F.51)
End of Proof.
Proof (of (F.48)).
For the proof of (F.48), we depart from the vertical mixed equiareal cylindric mapping (F.32), compute
the Jacobi matrix of partial derivatives of {x(Λ, Φ), y(Φ)} and constitute the coordinates of the left
Cauchy–Green deformation tensor c AB as well as the matrix C l G
−1
l
implementing (F.14) and finally
the trace tr
C l G
−1
l
.
x(Λ, Φ) =
(α + β)A cos Φ
α cos Φ + β
1 − E 2 sin
2 Φ
Λ ,
y(Φ) =
A(1 − E
2 )
α + β
⎛
⎝ β
Φ
0
dΦ
∗
(1 − E 2 sin
2 Φ ∗ ) 3/2 +
α
4E
ln
1 + E sin Φ
1 − E sin Φ
+
2E sin Φ
1 − E 2 sin
2 Φ
⎞
⎠ ,
(F.52)
x Λ =
A(α + β)L
αL + βA
, y Λ = 0 , x Φ = −
A
2 (α + β)β sin ΦM Λ
(αL + βA) 2
, y Φ =
M (αL + βA)
A(α + β)
, (F.53)
c 11 = x
2
Λ + y
2
Λ =
A
2 (α + β)
2 L
2
(αL + βA) 2 ,
c 12 = x Λ x Φ + y Λ y Φ = −
A
3 (α + β)
2 β sin ΦLM Λ
(αL + βA) 3
,
c 22 = M
2
A
4 (α + β)
2 β
2 sin
2 ΦΛ
2
(αL + βA) 4
+
(αL + βA)
2
A 2 (α + β) 2
,
(F.54)
det [c AB ] = det [G AB ] =
A
4 (1 − E
2 )
2 cos
2 Φ
(1 − E 2 sin
2 Φ) 4 ,
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
A
2 (α + β)
2
(αL + βA) 2 +
A
6 (α + β)
4 β
2 Λ
2 sin
2 Φ + (αL + βA)
6
A 2 (α + β) 2 (αL + βA) 4
.
(F.55)
End of Proof.