Geodesics - Map Projections: Cartographic Information Systems - page 742

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

1

2 ( p 1 + p 2 ) 2 ρ 2 (1 + d 02 ( q 2 ) 2 + d 12 q 1 ( q 2 ) 2 +

d 22 ( q 1 ) 2 ( q 2 ) 2 +

−

(E.95)

+ d 32 ( q 1 ) 3 ( q 2 ) 2 + d 04 ( q 2 ) 4 + d 14 q 1 ( q 2 ) 4 +O λ 2 (( q 1 ) 6 , ( q 2 ) 6 )) .

(iv)Universal Conic Projection (UC) (variant one variant two):
L 2 ( q ( t ) , q ( t )) = 1

2 ( q 1 ) 2 +( q 2 ) 2 +
A 2 cos 2 r − 1 (( q 1 ) 2 +( q 2 ) 2 )
2 c 2 n 2 1 e 2 sin 2 r 1 ( q 1 ) 2 +( q 2 ) 2 ×

(E.96)

−

r − 1 ( q 1 ) 2 +( q 2 ) 2

⎛

⎝ tan 2 ⎛

⎞

π
4 −

⎝ ⎠

×

2

1+ e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

⎛ ⎞ ⎞ e −n

⎝ ⎠ ⎠

e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

,

1 −

A 2 cos 2 r − 1 (( q 1 ) 2 +( q 2 ) 2 )
2 c 2 n 2 (1

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

2 ( p 1 + p 2 )

−

e 2 sin 2 r 1 ( ( q 1 ) 2 +( q 2 ) 2 )) ×

(E.97)

−

r − 1 ( q 1 ) 2 +( q 2 ) 2

⎛

⎝ tan 2 ⎛

⎞

π
4 −

⎝ ⎠

×

2

1+ e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

⎛ ⎞ ⎞ e −n

⎝ ⎠ ⎠

e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

.

1 −

2

Box E.6 (The differential equations of a geodesic in

A,B 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).

E

(i)Universal Polar Stereographic Projection (UPS):

A 2
( q 1 )+( q 2 ) 2

q μ +

(1 − e 2 )cos f − 1 ( q 1 ) 2 +( q 2 ) 2 sin f − 1 ( q 1 ) 2 +( q 2 ) 2
1

× (E.98)

e 2 sin 2 f − 1 ( q 1 ) 2 +( q 2 ) 2 2 −