Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 765

c 11 = x Λ + y Λ = ( αA + βL ) 2

β sin ΦM ( αA + βL ) Λ
( α + β ) 2

,c 12 = x Λ x Φ + y Λ y Φ = − ,

(F.58)

( α + β ) 2

c 22 = x Φ + y Φ = M 2 β 2 sin 2 ΦΛ 2

( αA + βL ) 2 ,
det[ c AB ]=det[ G AB ]= A 4 (1 − E 2 ) 2 cos 2 Φ
(1

+ ( α + β ) 2 L 2

( α + β ) 2

,

E 2 sin 2 Φ ) 4

−

G 22 = ( αA + βL ) 2 ( α + β ) 2 L 2 + β 2 Λ 2 sin 2 Φ ( αA + βL ) 2 +( α + β ) 4 L 2

tr[C l G − l ]= c 11

G 11 + c 22

. (F.59)

( α + β ) 2 ( αA + βL ) 2

End of Proof.

Box F.3 (Left Cauchy-Green deformation tensor (vertical mixed equiareal cylindric
mapping)).

The coordinates of the left Cauchy-Green deformation tensor
for the vertical mixed equiareal cylindric mapping:
c 11 = A 2 ( α + β ) 2 L 2

A 3 ( α + β ) 2 β sin ΦLMΛ
( αL + βA ) 3 ( αL + βA ) 2 , 12 = −

,

(F.60)

c 22 = M 2 A 6 ( α + β ) 4 β 2 sin 2 ΦΛ 2 +( αL + βA ) 6
A 2 ( α + β ) 2 ( αL + βA ) 4

.

Box F.4 (Left Cauchy-Green deformation tensor (horizontal mixed equiareal cylindric
mapping)).

The coordinates of the left Cauchy-Green deformation tensor
for the horizontal mixed equiareal cylindric mapping:
c 11 = ( αA + βL ) 2

β sin ΦM ( αA + βL ) Λ
( α + β ) 2

, 12 = − ,

( α + β ) 2

(F.61)

c 22 = M 2 β 2 Λ 2 sin 2 Φ ( αA + βL ) 2 +( α + β ) 4 L 2

.

( α + β ) 2 ( αA + βL ) 2

Corollary F.9 (Maximal left angular distortion of the vertical-horizontal mean of mixed equiareal
cylindric mapping of the biaxial ellipsoid

2

E

A,B ).

The right maximal angular distortion Ω generated by the mapping equations ( F.32 )forthe
vertical mean and ( F.43 )-( F.46 ) for the horizontal mean of mixed equiareal mappings of the
biaxial ellipsoid is represented by ( F.62 ) subject to ( F.63 ), where tr [C l G − l ] is given by ( F.48 )
for the vertical mean of ( F.32 )andby( F.49 ) for the horizontal mean of ( F.43 )-( F.46 ).