Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 635

Table 23.3 (continued)

5 th integrals

s
s 0

cos α 0 Δα 4 ds
=cos α 0

κ 0 s−s 0

κ 0 s−s 0 Δs 4 dΔs +3 κ 0 Δs 5 dΔs/ 2

0 0

Δs 8 dΔs/ 16 κ 0 s−s 0 Δs 6 dΔs/ 2+ κ 0 s−s 0 +3 κ 0

0 0

s
s 0

sin α 0 Δα 3 ds =
sin α 0

κ 0 s−s 0

Δs 4 dΔs +3 κ 0 κ 0 s−s 0 Δs 5 dΔs/ 2+

0 0

Δs 8 dΔs/ 16 3 κ 0 κ 0 s−s 0 Δs 6 dΔs/ 2+ κ 0 s−s 0

+ 4! s

0 0

cos α 0 Δα 4 ds
= κ 0 cos α 0 ( s

s 0

s 0 ) 5 / 120 + κ 0 κ 0 cos α 0 ( s s 0 ) 6 / 96 + κ 0 κ 0 cos α 0 − −

s 0 ) 7 / 112 + κ s 0 ) 9 / 3456 0 cos α 0 ( s

( s − −

+ 4! s sin α 0 Δα 4 ds

= κ 0 sin α 0 ( s

s 0

s 0 ) 5 / 120 + κ 0 κ 0 sin α 0 ( s s 0 ) 6 / 96 + κ 0 κ

0 sin α 0

− −

s 0 ) 7 / 112 κ 0 s 0 ) 9 / 3456

6 th integrals

( s − − sin α 0 ( s −

κ 0

s s−s 0 s−s 0

sin α 0 Δα 5 ds =sin α 0 Δs 5 dΔs +5 κ 0 κ 0 Δs 6 dΔs/ 2

s 0 0 0

s−s 0 s−s 0 s−s 0

+5 κ 0 κ 0 Δs 7 dΔs/ 2+5 κ 0 κ 0 Δs 8 dΔs/ 4+5 κ 0 κ 0

Δs 9 dΔs/ 16

0 0 0

Δs 10 dΔs/ 32

s−s 0

+ κ 0

0

κ 0

s s−s 0 s−s 0

cos α 0 Δα 5 ds =cos α 0 Δs 5 dΔs +5 κ 0 κ 0 Δs 6 dΔs/ 2

s 0 0 0

s−s 0 s−s 0

+5 κ 0 κ 0 Δs 7 dΔs/ 2+5 κ 0 κ 0

Δs 8 dΔs/ 4

0 0

s−s 0 s−s 0

Δs 10 dΔs/ 32 +5 κ 0 κ 0 Δs 9 dΔs/ 16 + κ 0

0 0

s

1
5!

s 0 ) 6 / 720 s 0 ) 7 / 336 sin α 0 Δα 5 ds = κ 0 sin α 0 ( s κ 0 κ 0 sin α 0 ( s − − − − − −

s 0

κ 0 κ 0 sin α 0 ( s s 0 ) 8 / 384 κ 0 κ 0 sin α 0 ( s s 0 ) 9 / 864 − − − −

κ 0 κ 0 κ 0 s 0 ) 10 / 3840 s 0 ) 11 / 42240 sin α 0 ( s − − sin α 0 ( s −

s

+ 1
5!

s 0 ) 6 / 720 + κ 0 κ 0 cos α 0 ( s s 0 ) 7 / 336 +

cos α 0 Δα 5 ds = κ 0 cos α 0 ( s

− −

s 0

κ 0 κ 0 cos α 0 ( s s 0 ) 8 / 384 + κ 0 κ 0 cos α 0 ( s s 0 ) 9 / 864 + − −

κ 0 κ 0 cos α 0 ( s − s 0 ) 10 / 3840 + κ 0 cos α 0 ( s − s 0 ) 11 / 42240