“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 360

s n := 1

1

n ! s ( n ) ( B 0 ) 2 . 1 s ( n ) ( B 0 ) , n ( n − 1) ·····

given in detail by the coecients of Box 15.1 .

Box 15.1 (Taylor expansion of r ( B )and s ( B )).

Taylor expansion of r ( B ) up to order three:
r : G 11 G 22 =
=cos B 1

E 2 sin 2 B
1 − E 2

−

=

(15.37)

N N

1

n ! r ( n ) ( B 0 ) b n =

r n b n , =

n =0 n =0

E 2 sin 2 B 0 )

r 0 =+ cos B 0 (1

−

,

1 − E 2

sin B 0 (1 + 2 E 2 3 E 2 sin 2 B 0 ) −

r 1 = − , (15.38)

1 − E 2

cos B 0 (1 + 2 E 2 9 E 2 sin 2 B 0 )

2(1 − E 2 )

−

r 2 = − ,

27 E 2 sin 2 B 0 )
6(1 − E 2 )

r 3 =+ sin B 0 (1 + 20 E 2

−

.

Taylor expansion of s ( B ) up to order three:
s := G 22 /G 11 =

E 2
cos B (1 − E 2 sin 2 B ) =

1 −

= (15.39)

N N

1

n ! s ( n ) ( B 0 ) b n =

s n b n , =

n =0 n =0

1 − E 2
cos B 0 (1

s 0 =

E 2 sin 2 B 0 ) ,

−

E 2 )sin B 0 (1 + 2 E 2 3 E 2 sin 2 B 0 )

s 1 = (1

− −

,

E 2 sin 2 B 0 ) 2 cos 2 B 0 (1 −