From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 81

E 2 A 1 ,A 1 ,A 2 , projective geometry of type central
perspective (perspective center S l , projection line S l P l p l , projective plane

Fig. 1.27. Left : vertical section of the “left” ellipsoid-of-revolution

P 2 N l ). Right : vertical section of the

S r , projective geometry of type central perspective (perspective center S r , projection line S r P rpr ,
projective plane

“right” sphere

P 2 N r )

2 1+ √ 1

,

= 1

− E 2 sin Φ 1

2 (1 + sin φ ) ,

√ 1 − E 2 sin 2 Φ

P 2 + Q 2 =2 A 1

cos Φ

1

E 2 sin 2 Φ + √ 1 − − E 2 +sin Φ

p 2 + q 2 =2 r

sin( 2 −

φ )
1+cos( 2 −

φ ) ,

p 2 + q 2 =2 r tan π P 2 + Q 2 =

2 A 1 φ

2

√ 1

E 2 ×

4 −

−

cos Φ

1 − E 2 sin 2 Φ/ √ 1 − E 2 +sin Φ

× , := g ( φ ) . (1.261)

P 2 + Q 2 =

2 A 1

√ 1

E 2 × −

cos Φ
1+sin Φ

1+sin Φ

1

× ,

E 2 sin 2 Φ/ √ 1 − − E 2 +sin Φ

E 2 tan π

P 2 + Q 2 =

2 A 1 φ

2

√ 1

4 −

×

−

(1 + E )(1

− E )(1 + sin Φ )

× (1 + E sin Φ )(1

E sin Φ )+ (1 + E )(1 E )sin Φ =: − −

=: f ( Φ ) .