Optimal Map Projections by Variational Calculus: Harmonic Maps - Map Projections: Cartographic Information Systems - page 563

Fig. 22.1. Reference Meridian Strip { L W ≤ L ≤ L E ,B S ≤ B ≤ B N } . Reference Longitude L 0 =( L W + L E ) / 2.

Symmetry: B S +B N =0

Table 22.3 Solution of the Laplace-Beltrami equation in terms of isometric (conformal, isothermal) coordinates

p,q :

E 2 A 1 ,A 2

E 2 A 1 ,A 2 : isometric coordinates of type Mercator ”

“

x ( Mercator )= A 1 L (22.34)

tan π

[ 1

4 + B

E sin B
1+ E sin B ] 2 }

−

y ( Mercator )= A 1 Q = A 1 ln { = A 1 lamB (22.35)

2

tan π

[ 1

4 + B

E sin B
1+ E sin B ] 2 }

−

Q ( B )= lamB := ln { (22.36)

2

A 1 cos lam − 1 Q/A 1 cosp/A 1
1

A 1 cos lam − 1 Q/A 1 sinp/A 1
1

X = E 1 + E 2 +

− E 2 sin 2 ( lam − 1 Q/A 1 ) − E 2 sin 2 ( lam − 1 Q/A 1 )

A 1 1 E 2 sin lam − 1 Q/A 1

−

+ E 3 1 (22.37)

− E 2 sin 2 ( lam − 1 Q/A 1 )

E 2 A 1 ,A 2

Left metric of

G LL = ∂ X

= ∂ X = G QQ

∂ X
∂L

∂ X
∂Q

∂L | ∂Q |

(22.38)

A 1 cos 2 lam − 1 Q 2

G LL ( Q )= G QQ ( Q )= G BB ( dQ

dB ) − 2 =

E 2 sin 2 ( lam − 1 Q ) := Λ 2 ( Q )

(22.39)

1 −

E 2 dQ

dB =

1 − 1

cosB

(22.40)

(1 − E 2 sin 2 B )