From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 78

Box 1.36 (The differential equation which governs the factor of conformality).

Two versions of the special Helmholtz equation ( k is the Gaussian curvature
k ( p, q )) :
(i) Δ ln λ 2 +2 kλ 2 =0 . (ii) Δλ 2 +2 kλ 4 =0 .

(1.253)

2

Right differential equation of the factor of conformality ( S

r ):

k r = 1

r 2 = constant ,Δ ln λ r + 2 r 2 λ r =0 ,

16 r 4

λ r =

(4 r 2 + p 2 + q 2 ) 2 , ln λ r =ln16 r 4

2ln(4 r 2 + p 2 + q 2 ) ,

−

p q

4 r 2 + p 2 + q 2 ,

D p ln λ r =

4 r 2 + p 2 + q 2 ,D q ln λ r =

− 4 − 4

4 4 r 2 + p 2 + q 2

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

−

D pp In λ r = D p ln λ r =

− ,

D qq In λ r = D q In λ r =

4 4 r 2 + p 2 + q 2

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

−

− , (1.254)

4

D pp In λ r =

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

p 2 + q 2 ) ,D qq In λ r = − −

4

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

q 2 ) ,

− −

32 r 2
(4 r 2 + p 2 + q 2 ) 2 =

2

Δ r ln λ r =

r 2 λ r q. e. d.

− −

2
A 1

Left differential equation of the factor of conformality ( E A 1 ,A 2 ):

E 2 sin 2 φ ) 2
A 1 (1 − E 2 )

k l = (1

−

=

E 2 sin 2 f − 1 ( P 2 + Q 2 )
A 1 (1

= 1

−

,

− E 2 )

Δ ln λ l +2 k ( P,Q ) λ l =0 ,

A 1 cos 2 f − 1 ( P 2 + Q 2 )

1
P 2 + Q 2 ,

λ l =

E 2 sin 2 f − 1 ( P 2 + Q 2 )

(1.255)

1 −

ln 1 E 2 sin 2 f − 1 ( P 2 + Q 2 )

ln λ l =ln A 1 + 2 lncos f − 1 ( P 2 + Q 2 )

− −

ln( P 2 + Q 2 ) ,
Δ l ln λ l =( D P + D Q )ln λ l =

−

2 k ( P,Q ) λ l

−

q. e. d.