Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 764

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

and constitute the coordinates
of the left Cauchy-Green deformation tensor c AB as well as the matrix C l G − i implementing ( F.14 )
and finally the trace tr [C l G − l ].

{ x ( Λ, Φ ) ,y ( Φ ) }

( α + β ) A cos Φ
α cos Φ + β 1 − E 2 sin 2 Φ

x ( Λ, Φ )= Λ,

⎛

⎞
⎠ , (F.52)

ln 1+ E sin Φ

Φ

E 2 )
α + β

d Φ ∗

y ( Φ )= A (1

− α

4 E

2 E sin Φ

⎝ β

E 2 sin 2 Φ ∗ ) 3 / 2 +

1 − E sin Φ +

E 2 sin 2 Φ

(1 − 1 −

0

x Λ = A ( α + β ) L

A 2 ( α + β ) β sin ΦMΛ
( αL + βA ) 2

,y Φ = M ( αL + βA )
A ( α + β ) αL + βA ,y Λ =0 ,x Φ = −

, (F.53)

c 11 = x Λ + y Λ = A 2 ( α + β ) 2 L 2

( αL + βA ) 2 ,

A 3 ( α + β ) 2 β sin ΦLMΛ
( αL + βA ) 3

c 12 = x Λ x Φ + y Λ y Φ = − , (F.54)

c 22 = M 2 A 4 ( α + β ) 2 β 2 sin 2 ΦΛ 2

A 2 ( α + β ) 2 ,

+ ( αL + βA ) 2

( αL + βA ) 4

det[ c AB ]=det[ G AB ]= A 4 (1

E 2 ) 2 cos 2 Φ

−

,

E 2 sin 2 Φ ) 4 (1 −

( αL + βA ) 2 + A 6 ( α + β ) 4 β 2 Λ 2 sin 2 Φ +( αL + βA ) 6

tr[C l G − l ]= c 11

G 11 + c 22 G 22 = A 2 ( α + β ) 2

.

(F.55)

A 2 ( α + β ) 2 ( αL + βA ) 4

End of Proof.

Proof (of ( F.49 )).

For the proof of ( F.49 ), we depart from the horizontal mixed equiareal cylindric mapping ( F.43 )-
( F.46 ), compute the Jacobi matrix of partial derivatives of and constitute the

coordinates of the left Cauchy-Green deformation tensor c AB as well as the matrix C l G − l imple-
menting ( F.14 ) and finally the trace tr [C l G − l ].

{ x ( Λ, Φ ) ,y ( Φ ) }

α + ,

AΛ
α + β

β cos Φ

1

x ( Λ, Φ )=

E 2 sin 2 Φ

−

Φ

cos Φ ∗ (1 E 2 sin 2 Φ ∗ ) − 3 / 2

−

E 2 )( α + β ) d Φ ∗ ,

y ( Φ )= A (1 −

α 1

(F.56)

E 2 sin 2 Φ ∗ + β cos Φ ∗ −

0

x Λ = αA + βL
α + β

β sin ΦMΛ
α + β

,y Φ = ( α + β ) LM

,y Λ =0 ,x Φ = −

αA + βL ,

(F.57)