Geodesics - Map Projections: Cartographic Information Systems - page 743

∂f − 1 ( q 1 ) 2 +( q 2 ) 2

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

× −

∂q μ

e 2 sin 2 f − 1 ( q 1 ) 2 +( q 2 ) 2 q μ =0 1 −

∀ μ =1 , 2 ,

q μ = δ μν p ν ∀

μ, ν =1 , 2 , (E.99)

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

p μ = −

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

(1 −

× (E.100)

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

−

∂f − 1 ( q 1 ) 2 +( q 2 ) 2

A 2
[( q 1 ) 2 +( q 2 ) 2 ] 2

× +

∂q μ

cos f − 1 ( q 1 ) 2 +( q 2 ) 2

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

∀ μ =1 , 2 .

(ii)Universal Mercator Projection (UPS):

(1 − e 2 )sin f − 1 ( q 2 /A )

cos 3 f − 1 ( q 2 /A )

∂f − 1 ( q 2 /A )
∂q 2

q 1 =0 , q 2

− =0 , (E.101)

q μ = δ μν p ν ,

(E.102)

p 1 =0( p 1 cyclic) ,p 1 =const ., (E.103)

e 2 )sin f − 1 ( q 2 /A )
cos 3 f − 1 ( q 2 /A )

∂f − 1 ( q 2 /A )
∂q 2

p 2 = (1

−

. (E.104)

(iii)Universal Transverse Mercator Projection (UTM):

1

q 1

2 ρ 2 [ d 12 ( q 2 ) 2 +2 d 22 q 1 ( q 2 ) 2 +3 d 32 ( q 1 ) 2 ( q 2 ) 2 + d 14 ( q 2 ) 4 +O λ 2 (( q 1 ) 5 ,
( q 2 ) 6 )] = 0 ,

−