From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 80

Here, we conclude with a representation of the left as well as the right inverse mapping, namely
Φ − l :

}→ X ( P,Q )and Φ − r :

x ( p, q ) in terms of conformal coordinates (isometric,
isothermal) of Box 1.37 , which specializes Φ − l and Φ − r of Box 1.21 .

{ P, Q { p, q }→

End of Solution (the fourth problem).

Solution (the fifth problem).

By means of Fig. 1.27 , we illustrate why “UPS” is called stereographic . The stereographic projec-
tion of the “left” ellipsoid-of-revolution

r isbaseduponthree
elements of projective geometry of type central perspective . First, we define the perspective center ,
here the ellipsoidal “left” South Pole S l as well as the spherical “right” South Pole S r . Second, we
define the bundle of projection lines leaving S l and S r , respectively, and intersecting

A 1 ,A 1 ,A 2

E and the “right” sphere S

2
A 1 ,A 1 ,A 2

E at P l

2 2

N l

2
N r

and S

r at S r . Third, we define the projective plane

P and P , respectively, namely the tangent

A 1 ,A 1 ,A 2 r at the “right” spherical North

planes T N l E at the “left” ellipsoidal North Pole and T N r S

Pole, respectively. The projection lines S l → P l intersect the projective plane at p l ,anelement

of the “left” tangent plane at the “left” North Pole, and the projection lines S r

P r intersect
the projective plane at p r , an element of the “right” tangent plane at the “right” North Pole.
Note that we have collected th e fundam ental “left” and “rig ht” ratios of projective geometry in

→

Box 1.38 . Their conversion to P 2 + Q 2 “left” and p 2 + q 2 “right” generates the map Φ

→ f ( Φ )

and φ → g ( φ ), respective ly. The projective planes are covered by po lar coord inates of type

{ P 2 + Q 2 cos α l , P 2 + Q 2 sin α l } { p 2 + q 2 cos α r , p 2 + q 2 sin α r } “left” and of type “right” ,

P 2 + Q 2 , p 2 + q 2

respectively. { } are the radial coordinates , { α l ,α r } are the “le ft” and “right”

South azimuths . The central perspective generates P 2 + Q 2 = f ( Φ )versus p 2 + q 2 = g ( φ )
and α l = Λ versus α r = λ . Indeed, “UPS” is azimuth preserving: the “left” azimuth is identified
as ellipsoidal longitude, the “right” azimuth as spherical longitude, and

P ( Λ, Φ )= f ( Φ )cos Λ versus p ( λ, φ )= g ( Φ )cos λ,

(1.259)

Q ( Λ, Φ )= f ( Φ )sin Λ versus q ( λ, ( p )= g ( Φ )sin λ.

End of Solution (the fifth problem).

Box 1.38 (Projective geometry of type central perspective).

Left projective ratio: Right projective ratio:

S l N l , √ X 2 + Y 2

Z l P l

N l p l = S l Z l

P 2 + Q 2 = A 2 + Z

Z r P r

N r p r = S r Z r

.

S r N r ,

2 A 2

x 2 + y 2

p 2 + q 2 = r + z

. (1.260)

2 r

f ( Φ ): g ( φ ):

A 1 cos Φ
1 − E 2 sin 2 Φ

1
P 2 + Q 2 =

r cos φ
p 2 + q 2 =