Geodesics - Map Projections: Cartographic Information Systems - page 744

1

q 2

2 ρ 2 [2 d 02 q 2 +2 d 12 q 1 q 2 +2 d 22 ( q 1 ) 2 q 2 +2 d 32 ( q 1 ) 3 q 2 +4 d 04 ( q 2 ) 3 +

− (E.105)

+4 d 14 q 1 ( q 2 ) 3 + O λ 2 (( q 1 ) 6 , ( q 2 ) 5 )] = 0 ,

q μ = δ μν p ν ,

(E.106)

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 )] ,

p 1 =

1

2 ρ 2 [2 d 02 q 2 +2 d 12 q 1 q 2 +2 d 22 ( q 1 ) 2 q 2 +2 d 32 ( q 1 ) 3 q 2 +4 d 04 ( q 2 ) 3 +

p 2 = (E.107)

+4 d 14 q 1 ( q 2 ) 3 +O λ 2 (( q ⊥ ) 6 , ( q 2 ) 5 )] .

(iv) Universal Conic Projection (UC) (variant one and variant two):

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

⎡

A 2 (1

−

⎣

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

q μ −

− −

r − 1 ( q 1 ) 2 +( q 2 ) 2

⎞

× tan π

⎠

4 − 2

2 ⎞
⎠

1+ e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

⎞

− 2 n− 1

e

⎠

e sin r − 1 ( q 1 ) 2 +( q 2 ) 2

− (E.108)

1 −

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

A 2 (1

−

− ×

− 2 n ⎤
⎦ ×

2 ⎞
⎠

r − 1 ( q 1 ) 2 +( q 2 ) 2
2

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

⎞ ⎞

e

tan π

⎠ ⎠

×

4 −

∂r − 1 ( q 1 ) 2 +( q 2 ) 2

× =0 ,

∂q μ

q μ = δ μν p ν ,

(E.109)