From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 79

Box 1.37 (Representation of Φ − l and Φ − r in terms of conformal coordinates: E

A 1 ,A 1 ,A 2 → S r ) .

Φ − l : X ( Λ, Φ )= Φ − r : x ( λ, φ )=

= E 1 A 1 cos Φ cos Λ 1 + = e 1 r cos φ cos λ +

E 2 sin 2 Φ

−

+ E 2 A 1 cos Φ sin Λ

1 − E 2 sin 2 Φ + + e 2 r cos φ sin λ +

E 2 )sin Φ
1 − E 2 s in 2 Φ + E 3 A 1 (1

−

= + e 3 r sin φ =

A 1 cos f − 1 ( P 2 + Q 2 ) P p

4 r 2 + p 2 + q 2 +

P 2 + Q 2 += e 1 4 r 2 = E 1 1 (1.256)

E 2 sin 2 f − 1 P 2 + Q 2 −

A 1 cos f − 1 ( P 2 + Q 2 ) Q q

4 r 2 + p 2 + q 2 +

P 2 + Q 2 ++ e 2 4 r 2

= E 2 1 − E 2 sin 2 f − 1 P 2 + Q 2

E 2 )sin f − 1 ( P 2 + Q 2 )

+ e 3 r 4 r 2

( p 2 + q 2 )
4 r 2 + p 2 + q 2 .

+ E 3 A 1 (1

− −

1

.

E 2 sin 2 f − 1 ( P 2 + Q 2 ) −

Isoparametric mapping:
p = P, q = Q. (1.257)

Solution (the third problem).

By means of the two mapping equations “left” { P = f ( Φ )cos Λ, Q = f ( Φ )sin Λ } and of the two

mapping equations “right” { p = g ( φ )cos λ, q = g ( φ )sin λ } , we are able to compute the factors

Λ l ,λ l } Λ r ,λ r } of type “right”. The detailed formulae are

reviewed in Box 1.34 . If we specify “left” Φ = π/ 2 (ellipsoidal North Pole) or “right” φ = π/ 2
(spherical North Pole), we are led to l Λ 1 ( π/ 2) = l Λ 2 ( π/ 2) = Λ l ( π/ 2) = 1 and r Λ 1 ( π/ 2) =
r Λ 2 ( π/ 2) = Λ r ( π/ 2) = 1. Obviously, at the North Pole, “left UPS” and “right UPS” are an
isometry. We shall see later that this is a built-in constraint for any UPS.

of conformality { of type “left” and {

End of Solution (the third problem).

Solution (the fourth problem).

A “simple conformal mapping” of E

A 1 ,A 1 ,A 2 → S r is the isoparametric mapping, which is conve-

niently characterized by
p = P, q = Q or 2 r tan 4 −

2 cos λ = 2 ) 1+ E sin Φ 1 −E sin Φ E/ 2 cos Λ,

(1 − E ) E/ 2 φ

2 A 1
√ 1 −E 2

(1+ E ) E/ 2 tan( 4 −

Φ

(1.258)

2 r tan 4 −

2 sin λ = 2 ) 1+ E sin Φ 1 −E sin Φ E/ 2 sin Λ.

(1 − E ) E/ 2

φ 2 A 1

(1+ E ) E/ 2 tan( 4 −

Φ

√ 1 −E 2