Geodesics - Map Projections: Cartographic Information Systems - page 741

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

d s 2 =

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

(E.89)

−

r − 1 ( q 1 ) 2 +( q 2 ) 2

⎛

⎝ tan 2 ⎛

⎞

π
4 −

⎝ ⎠

×

2

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

⎛ ⎞ ⎞

−n

e

( q 1 ) 2 +( q 2 ) 2 .

⎝ ⎠ ⎠

( λ ( q 1 .q 2 ) from series inversion of q 1 ( Λ, Φ )and q 2 ( Λ, Φ ) leading to Λ ( q 1 ,q 2 )and Φ ( q 1 ,q 2 ).
See Snyder ( 1987a , p. 109).

Box E.5 (The Lagrangean version versus the Hamiltonian version of a geodesic in E

2 A,B in

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

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

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

A 2 cos 2 f − 1 ( q 1 ) 2 +( q 2 ) 2
2 1

1
( q 1 ) 2 +( q 2 ) 2 ,

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

+ (E.90)

−

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

2 ( p 1 + p 2 )

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

1
( q 1 ) 2 +( q 2 ) 2 .

− (E.91)

(ii)Universal Mercator Projection (UM):
L 2 ( q ( t ) , q ( t )) = 1

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

e 2 sin 2 f 1 ( q 2 /A )
2cos 2 f − 1 ( q 2 /A )

−

, (E.92)

e 2 sin 2 f − 1 ( q 2 /A )
2cos 2 f 1 ( q 2 /A )

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

1 −

2 ( p 1 + p 2 )

− . (E.93)

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

2 [( q 1 ) 2 +( q 2 ) 2 ]+ 1 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.94)

+ 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 )) ,