Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 768

C 22 = G 11 Λ y + G 22 Φ y ,
C 11 = N 2 cos 2 Φ

,

x Λ

N 2 cos 2 Φx Φ
x Λ y Φ

C 12 = − , (F.71)

C 22 = N 2 cos 2 Φx Φ + M 2

y Φ . x Λ y Φ

The coordinates of the right Cauchy-Green deformation tensor for the vertical as well as the hor-
izontal mixed equiareal cylindric mapping of ( F.32 )and( F.43 )-( F.46 ) are collected in Boxes F.5
and F.6 . The results enable us to compute the right eigenvectors given by Corollary F.11 . Indeed,
they are needed to orientate by tan ϕ = C 12 / ( λ 2 −

C 11 ) the right principal stretches, once we
relate Λ 1 = λ − 1 such that λ 1 = Λ − 1 and λ 2 = Λ − 2 holds. There exists a right analogue ω of the
left maximal angular distortion Ω ,namelyof( F.62 ), as soon as we replace left principal stretches
by right ones.

Box F.5 (The coordinates of the right Cauchy-Green deformation tensor for the vertical
mixed equiareal cylindric mapping of ( F.32 )).
The coordinates of the right Cauchy-Green deformation tensor for the vertical mixed equiareal
cylindric mapping of ( F.32 )areprovidedby

C 11 = ( αL + βA ) 2

A 2 ( α + β ) 2 , C 12 = Aβ sin Φ αL + βA ,

(F.72)

( αL + βA ) 2 1+ ( A 2 β 2 sin 2 Φ ( αL + βA ) 2 .

C 22 = A 2 ( α + β ) 2

Box F.6 (The coordinates of the right Cauchy-Green deformation tensor for the horizontal
mixed equiareal cylindric mapping of ( F.43 )-( F.46 )).
The coordinates of the right Cauchy-Green deformation tensor for the horizontal mixed
equiareal cylindric mapping of ( F.43 )-( F.46 )areprovidedby

C 11 = ( α + β ) 2 L 2

( αA + βL ) 2 ,C 12 = β sin ΦLΛ αA + βL ,

(F.73)

( α + β ) 2 β 2 sin 2 ΦΛ 2 + ( αA + βL ) 2

.

1

C 22 =

L 2

Corollary F.11 (The right eigenvectors of the right Cauchy-Green deformation tensor).