From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 89

2 2

Box 1.40 (Left Cauchy-Green matrix, left eigenspace: E

A 1 ,A 1 ,A 2 → S r ).

Left manifold ( { Λ, Φ } coordinates) : Right manifold( { λ, φ } coordinates) :

A 1 cos 2 Φ Λ 1 (1 E 2 ) 2 −

d S 2 =

E 2 sin 2 Φ d Λ 2 + E 2 sin 2 Φ ) 3 d Φ 2 . d s 2 = r 2 cos 2 φ d λ 2 + r 2 d φ 2

(1.273)

1 − (1 −

“Ansatz”:

2 2

E

A 1 A 1 ,A 2 → S r ; λ = Λ, φ = f ( Φ ) .

(1.274)

Left Cauchy-Green matrix:
C l =J l G r J l = r 2 cos 2 φ

,

0

r 2 f 2 ( Φ ) 0

J l = D Λ λD Λ φ

= 1O , G r = r 2 cos 2 φ 0

r 2 .

(1.275)

0 f ( Φ )

D Φ λD Φ φ 0

Left principal stretches, left eigenspace:
| C l − Λ l G l | =

c 11 − G 11 Λ 2 0

⇔

=0

0 c 22 − G 22 Λ l

Λ 1 = c 11 .

c 11

= r 2 cos 2 φ

E 2 sin 2 Φ )

A 1 cos 2 Φ (1

−

G 11 Λ l =0
c 22 − G 22 Λ l =0

−

G 11

⇔ ⇔ (1.276)

r 2 f 2 ( Φ )
A 1

Λ 2 = c 22 E 2 sin 2 Φ ) 3 =

(1 −E 2 ) 2 (1

−

G 22

Box 1.41 (Equiareal mapping: E

A 1 ,A 1 ,A 2 → S r ,φ = f ( Φ )).

Area preserving postulate:

A 1 cos Φ 1

E 2 sin 2 Φ rf Φ
A 1 (1

r cos φ

E 2 sin 2 Φ ) 3 / 2 =1 . Λ 1 Λ 2 =1 ⇔ −

E 2 ) (1

− (1.277)

−

Equation of variables:

d Φ

r 2 cos φ d φ =

E 2 sin 2 Φ ) 2 A 1 (1 − E 2 )cos Φ.

(1.278)

(1 −

Standard integrals:
Δ := π/ 2 − Φ ⇒− d Δ =d Φ,

cos Φ sin Δ

(1 − E 2 cos 2 Δ ) 2 d Δ =

E 2 sin 2 Φ ) 2 d Φ =

−

(1 −

cos Δ

4 E ln 1+ E cos Δ

1

=

E 2 cos 2 Δ ) + E cos Δ + c r =

(1.279)

2(1 − 1 −

sin Φ

4 E ln 1+ E sin Φ

1

=

E 2 sin 2 Φ ) + E sin Φ + c r , 2(1 − 1 −