“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 359

Q 0 ( l,b ):= y 0 ,
Q l ( l,b ):= y 10 l + y 01 b =
α + β =1

y αβ l α b β ,

Q 2 ( l,b ):= y 20 l 2 + y 11 lb + y 02 b 2 =
α + β =2

y αβ l α b β , (15.32)

.

Q n ( l,b ):=
α + β = n

y αβ l α b β ,

subject to the Taylor expansion

r := G 11 G 22 =cos B 1

E 2 sin 2 B
1 − E 2

−

= r 0 + r 1 b + r 2 b 2 + r 3 b 3 +O(4) ,

(15.33)

namely

r 0 := 1

0! r (0) ( B 0 )= r ( B 0 ) ,
r 1 := 1

1! r (1) ( B 0 )= r ( B 0 ) ,

(15.34)

.
r n := 1

1

n ! r ( n ) ( B 0 )= 1 r ( n ) ( B 0) ,

n ( n − 1) ···· 2 ·

and vice versa

s := G 22 G 11 =

1 − E 2

1
cos B E 2 sin 2 B = s 0 + s 1 b 2 + s 3 b 3 +O(4)= 1 −

(15.35)

= r − 1 r − 0 r 1 b +( r − 0 r 1 − r − 0 r 2 ) b 2 +( r − 0 r 1 +2 r − 0 r 1 r 2 − r − 0 r 3 ) b 2 +O(4) ,

− −

0

namely

s 0 := 1

0! s (0) ( B 0 )= s ( B 0 ) ,
s 1 := 1

1! s (1) ( B 0 )= s ( B 0 ) ,

(15.36)

.