From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 108

1st mapping:

λ
φ

= Λ/ 2 , a = b = R.

(1.344)

Φ

2nd mapping (oblique orthogonal projection).

a,b :
X = R (1 + cos Φ )cos Λ/ 2 ,
Y = R (1 + cos Φ )sin Λ/ 2 ,
Z = R sin Φ ;

Left manifold T

2
O

Right manifold (oblique plane) P :

x = Y ,
y = Z ;

(1.345)

⎡ ⎤ ⎡ ⎤

X
Y
Z

X
Y
Z

⎣ ⎦ =R 2 ( β ) ⎣ ⎦

⇔

⎡ ⎤ ⎡ ⎤

X
Y
Z

X
Y
Z

⎣ ⎦ =R 2 ( −β ) ⎣ ⎦ ,

(1.346)

⎡ ⎤

cos β 0 sin β

01 0
sin β 0sin β

−

⎣ ⎦

R 2 ( β )=

⇔

⎡ ⎤

cos β 0sin β
010

⎣ ⎦ =R 2 ( β )=R 2 I(

− β );

− sin β 0cos β

x := Y = Y ,y := Z = − sin βX +cos βZ,

(1.347)

x = R (1 + cos Φ )sin Λ/ 2 ,y = −R (1 + cos Φ )sin β cos Λ/ 2+ R cos β sin Φ.

Box 1.59 (Left principal stretches).

Left and right matrices of the metric:

G l := R 2

R 2 ,

cos 2 Φ 0

0

G r := 10

.

(1.348)

01

Left Jacobi matrix: