“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 367

[( 15.72 )] y 22 3 [( 15.72 )] x 22 =0 ,

2 r 0 x 13

1
2 r 2 x 11 ,

− r 1 x 12 −

[( 15.72 )] x 13 = 1

s 0 (4 y 04 −

s 1 x 12 − s 2 x 11 − s 3 x 10 ) . [( 15.72 )] y 13 =0 .

(15.83)

Theorem 15.4 (The solution of the Korn-Lichtenstein equations of conformal mapping which
generates directly Gauss-Krueger or UTM conformal coordinates).

The equidistant mapping of the meridian of reference L 0 , which is the constraint fixing the general
solution ( 15.84 ) of the Korn-Lichtenstein equations ( 15.85 ) subject to the integrability conditions,
the Laplace-Beltrami equations given by ( 15.86 ), leads us to the solution ( 15.87 ) in the function
space of bivariate polynomials (Figs. 15.3 and 15.4 ).

x ( l,b )=0 , (0 ,b )= ∞ y 0 n b n ,

(15.84)

n =1

G 11 /G 22 y b =0 , b G 22 /G 11 y l =0 ,

x l − − (15.85)

Δ LB x ( l,b )

y ( l,b ) =0 ,

(15.86)

x ( l,b )=
= x 10 l + x 11 lb + x 30 l 3 + x 12 lb 2 + x 31 l 3 b + x 13 lb 3 + x 50 l 5 + x 32 l 3 b 2 +
+ x 14 lb 4 +O(6) , (15.87)

y ( l,b )=
= y 01 b + y 20 l 2 + y 02 b 2 + y 21 l 2 b + y 03 b 3 + y 40 l 4 + y 22 l 2 b 2 + y 04 b 4 + y 41 l 4 b + y 23 l 2 b 3 +
+ y 05 b 5 +O(6) .
Boxes 15.4 and 15.5 are a collection of the coecients x 10 , ..., y 05 .

End of Theorem.

Box 15.4 (A representation of the non-vanishing coecients in a polynomial setup of a
conformal mapping of type Gauss-Krueger or UTM).

x ( l,b )=
= x 10 l + x 11 lb + x 30 l 3 + x 12 lb 2 + x 31 l 3 b + x 13 lb 3 + x 50 l 5 + x 32 l 3 b 2 + (15.88)

+ x 14 lb 4 +O(6) .

A 1 cos B 0

x 10 =

E 2 sin 2 B 0 ) 1 / 2 , (1 −