“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 397

+O 3 s ,

y 1
ρ −

y 0 + s 02 y 1 y 0 2

+ s 11 x 1
ρ ρ −

y 2 = y 1 + ρ t 00 + t 10 x 1

ρ + t 01 y 1 ρ − y 0 + t 20 x 1

2

+

ρ

(15.124)

+O 3 t .

y 1

y 0 + t 02 y 1 y 0 3

+ t 11 x 1

ρ − ρ − ρ

The coecients are collected in Box 15.12 and are computed by MATHEMATICA up to

order five as a polynomial in terms of L 01 −

{ s ij ,t ij }

L 02 . Figure 15.19 outlines the transformation steps
of various orders. It should be noted that the explicit formula manipulation of the coecients
{ s ij ,t ij } in Box 15.12 was restricted by the postulate x 1 = x 2 and y 1 = y 2 if L 01 − L 02 =0holds!

Box 15.12 (Conformal polynomial { x 2 = X ( x 1 ,y 1 ) ,y 2 = Y ( x 1 ,y 1 ) } , coecients { s ij ,t ij } ).

s 00 = A 1 ( L 01 − L 02 )cos B 0
(1

E 2 sin 2 B 0 ) 1 / 2

−

L 02 ) 3 cos B 0 (1 2sin 2 B 0 + E 2 sin 4 B 0 )

+ A 1 ( L 01 −

−

+

E 2 sin 2 B 0 ) 1 / 2

6(1 − E 2 )(1 −

L 02 ) 5 cos B 0 [5 E 2 28 sin 2 B 0 16 E 2 sin 2 B 0 +24sin 4 B 0

+ A 1 ( L 01 − − − −

+86 E 2 sin 4 B 0 +

+26 E 4 sin 4 B 0 − 72 E 2 sin 6 B 0 − 100 E 4 sin 6 B 0 − 12 E 6 sin 6 B 0

+77 E 4 sin 8 B 0 +39 E 6 sin 8 B 0 −

28 E 6 sin 10 B 0 ] 120(1 E 2 sin 2 B 0 ) 1 / 2 − 1 , E 2 ) 3 (1 − − −

L 02 ) 2 (1 2sin 2 B 0 + E 2 sin 4 B 0 )

2(1

s 10 = ( L 01

− −

+

− E 2 )

16 E 2 sin 2 B 0 +24sin 4 B 0
+86 E 2 sin 4 B 0 +
+26 E 4 sin 4 B 0 − 72 E 2 sin 6 B 0 − 100 E 4 sin 6 B 0 − 12 E 6 sin 6 B 0
+77 E 4 sin 8 B 0 +39 E 6 sin 8 B 0

L 02 ) 4 [5 E 2 28 sin 2 B 0 −

+( L 01 − − −

−

28 E 6 sin 10 B 0 ][24(1 E 2 ) 3 ] − 1 ,

− −

s 01 = − ( L 01 − L 02 )sin B 0 −

L 02 ) 3 sin B 0 [5 E 2 6sin 2 B 0 − 6 E 2 sin 2 B 0 +9 E 2 sin 4 B 0 + − ( L 01 − − − (15.125)