From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 109

J l ;= D Λ xD Φ x

=

D Λ yD Φ y

= R

.

(1.349)

1

2 (1 + cos Φ )cos Λ/ 2

− sin Φ sin Λ/ 2

1

2 (1 + cos Φ )sin β sin Λ/ 2 s β cos Φ +sin β sin Φ cos Λ/ 2

Left Cauchy-Green matrix:

C l := J l G r J l = c 11 c 12

,

c 12 c 22

c 11 = x 4 + y A =

= 1

4 R 2 (1 + cos Φ ) 2 (cos 2 Λ/ 2+sin 2 β sin 2 Λ/ 2) =
= 1

4 R 2 (1 + cos Φ ) 2 (1

cos 2 β sin 2 Λ/ 2) ,

−

c 12 = x Λ x Φ + y Λ y Φ = (1.350)

= 1

2 R 2 (1 + cos Φ )sin Λ/ 2[

− sin Φ cos Λ/ 2

+sin β (cos β cos Φ +sin Φ sin β cos Λ/ 2)]

= 1

2 R 2 (1 + cos Φ )sin Λ/ 2(sin β cos β cos Φ

sin Φ cos 2 β cos Λ/ 2)

−

= 1

2 R 2 (1 + cos Φ )sin Λ/ 2cos β (sin β cos Φ

− sin Φ cos β cos Λ/ 2) ,

c 22 = x Φ + y Φ =
= R 2 [sin 2 Φ sin 2 Λ/ 2+(cos β cos Φ +sin Φ sin β cos Λ/ 2) 2 ] ,
det[C l ]= R 4 (1 + cos Φ ) 2 (sin β 1 sin Φ +cos Φ cos β cos Λ/ 2) 2 ,

(1.351)

4

det[G l ]= B 4 cos 2 Φ =det[C l ] .
Left principal stretches:
|

Λ l G l

C l − | =0 . (1.352)

With this box, we finish the general consideration of mappings between Riemann manifolds. In
the following chapter, we specialize the various rules for mappings between Riemann manifolds
and Euclidean manifolds.