“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 396

L 02 ) 2 +2 y 20 ( L 01 L 02 ) l 1 + y 20 l 1 + y 02 b 1 +

y 2 = ρ [ y 0 + y 01 b 1 + y 20 ( L 01 − − (15.120)

L 02 ) 2 b 1 +2 y 21 ( L 01 − L 02 ) l 1 b 1 + y 21 l 1 b 1 + y 03 b 1 +O 4 y ] .

+ y 21 ( L 01 −

Obviously, the conformal coordinates { x 2 ,y 2 } in the second strip L 02 depend on the difference

L 01 − L 02 of the chosen L 01 -strip, respectively. Finally, we have to replace { l 1 ,b 1 } within { x 2 ,y 2 }

by the bivariate homogeneous polynomial { l 1 ( x 1 ,y 1 ) ,b ( x 1 ,y 1 ) } given by ( 15.112 )and( 15.113 )

and coecients of Box 15.11 . In this way, we have achieved a solution of the strip

transformation problem presented in the form

{ l ij ,b ij }

x 2 = ρ x 10 ( L 02 − L 02 )+ x 10 l 10 x 1

y 1

y 0 +O 3 l +

ρ + l 11 x 1 ρ − ρ

+ x 11 ( L 01 − L 02 ) b 01 y 1 +0 3 b +

ρ − y 0 + b 20 x 1

2

+ b 02 y 1

ρ − y 0 2 ρ

+ x 11 l 10 x 1

y 1
ρ −

y 0 +O 3 l

ρ + l 11 x 1

× (15.121)

ρ

b 01 y 1 +O 3 x ,

y 0 + b 20 x 1
ρ

2

+ b 02 y 1 y 0 2 ×

ρ − ρ −

+0 3 b

y 2 = ρ y 0 + y 01 b 01 y 1 +O 3 b + y 0 + b 20 x 1

ρ

2

+ b 02 y 1 y 0 2

ρ − ρ −

L 02 ) l 10 x 1

y 1
ρ −

y 0 +O 3 l +

ρ + l 11 x 1

L 02 ) 2 +2 y 20 ( L 01

+ y 20 ( L 01 − −

ρ

+ y 20 l 10 x 1

y 1

ρ − y 0 +O 3 l 2

ρ + l 11 x 1

+ (15.122)

ρ

⎞

+ y 02 b 01 y 1 +O 3 b 2

ρ − y 0 + b 20 x 1

2

+ b 02 y 1

ρ − y 0 2

⎠ .

+O 3 y

ρ

In general:

x 2 = x 1 + ρ s 00 + s 10 x 1

ρ + s 01 y 1 ρ − y 0 + s 20 x 1

2

+

ρ

(15.123)