Geodesics - Map Projections: Cartographic Information Systems - page 734

n 2 sin 2 arctan 1 c ( q 1 ) 2 +( q 2 ) 2 1 /n

1

d s 2 = [(d q 1 ) 2 +(d q 2 ) 2 ] . (E.59)

2

Box E.2 (The Lagrangean version versus the Hamiltonian version of a geodesic in S

R in terms

of conformal coordinates (isometric coordinates) and Maupertuis gauge).

(i) Universal Polar Stereographic Projection (UPS):
L 2 ( q ( t ) , q ( t )) = 1

8 R 4
(4 R 2 +( q 1 ) 2 +( q 2 ) 2 ) 2 , 2 δ μν q μ q ν +

(E.60)

8 R 4
(4 R 2 +( q 1 ) 2 +( q 2 ) 2 ) 2 .
(ii) Universal Mercator Projection (UM):
L 2 ( q ( t ) , q ( t )) = 1

H 2 ( q ( t ) ,p ( t )) = 1

2 ( p 1 + p 2 ) −

2cosh 2 ( q 2 /R ) ,H 2 ( q ( t ) ,p ( t )) = 1

1

2 δ μν q μ q ν + 2 ( p 1 + p 2 )

1
2cosh 2 ( q 2 /R ) .

− (E.61)

(iii) Universal Transverse Mercator Projection (UTM):
L 2 ( q ( t ) , q ( t )) = 1

2 δ μν q μ q ν + 1 2 cos 2 arcsin[tanh( q 2 /R )] ,

(E.62)

H 2 ( q ( t ) ,p ( t )) = 1

1

2 cos 2 arcsin[tanh( q 2 /R )] .
(iv) Universal Conic Projection (UC):
L 2 ( q ( t ) , q ( t )) 1

2 ( p 1 + p 2 )

−

2 n sin 2 arctan 1 c ( q 1 ) 2 +( q 2 ) 2 1 /n

1

2 δ μν q μ q ν +

, (E.63)

2 n sin 2 arctan 1 c ( q 1 ) 2 +( q 2 ) 2 1 /n

H 2 ( q ( t ) ,p ( t )) = 1

1

2 ( p 1 + p 2 ) −

.

2

Box E.3 (The differential equations of a geodesic in

R in terms of conformal coordinates
(isometric coordinates) and Maupertuis gauge: Lagrange portrait, two differential equations
of second order, Hamilton portrait, four differential equations of first order).
(i) Universal Polar Stereographic Projection (UPS):

S

32 R 4

q μ +

(4 R 2 +( q 1 ) 2 +( q 2 ) 2 ) 3 q μ =0

∀ μ =1 , 2 ,

q μ = δ μν p ν ∀ μ, ν =1 , 2 (E.64)

(summation convention),