From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 67

Left Cauchy-Green matrix: Right Cauchy-Green matrix:

Λ l G l | Λ r G r |

| C l − =0 ⇔ | C r − =0 ⇔

r 2 Λ r g 11

A 1 − Λ l G 11 0 − 0

⇔

=0 ⇔⇔ =0 ⇔

q φ − Λ r g 22

0 Q Φ − Λ l G 22 0

l Λ 1 = A 1 r Λ 1 = r 2

= 1 − E 2 sin 2 Φ

(1.216)

g 11
r Λ 2 = q φ

cos 2 Φ G 11

⇔ ⇒⇔ ⇒

l Λ 2 = Q Φ

= 1 − E 2 sin 2 Φ

1
cos 2 φ

=

cos 2 Φ

g 22

G 22

⇒ l Λ 1 = l Λ 2 = Λ l = 1 − E 2 sin 2 Φ r Λ 1 = r Λ 2 = Λ r =

1
cos 2 Φ

⇔⇒ ⇔

cos 2 Φ

cos 2 Φ

⇔ λ l =

1 −E 2 sin 2 Φ .

⇔ λ r =cos 2 φ.

Box 1.28 (Representation of the factors of conformality in terms of conformal coordinates).

Left factor of conformality:
P = A 1 Λ, Q = A 1 f ( Φ ) ,f ( Φ ):=ln tan π

E/ 2 , 1 − E sin Φ

1+ E sin Φ 4 + Φ 2

E 2 sin 2 Φ
cos 2 Φ

cos 2 Φ

E 2 sin 2 Φ ,Λ l = 1

−

λ l =

(1.217)

1 −

⇒

cos 2 f − 1 ( Q/A 1 ) E 2 sin 2 f − 1 ( Q/A 1 )

cos 2 f − 1 ( Q/A 1 )

E 2 sin 2 f − 1 ( Q/A 1 ) , Λ l = 1

−

λ l =

1

.

−

Right factor of conformality:
p = rλ, q = r ln tan π

= r artanh sin φ,

4 + φ

2

1
cosh( q/r ) =cos φ,

tanh( q/r )=sin φ,

1
cos 2 φ

λ r =cos 2 φ, Λ r = (1.218)

⇒

1

cosh 2 ( q/r ) , Λ r =cosh 2 ( q/r ) .

λ r =

Box 1.29 (The differential equation which governs the factor of conformality).

Two versions of the special Helmholtz equations:
(i) Δ ln λ 2 +2 kλ 2 =0 , (ii) Δλ 2 +2 kλ 4 =0 .

(1.219)

( k is the Gaussian curvature k ( p, q ) . )
Right differential equation of the factor of conformality (

2

S

r ):