“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 364

Corollary 15.2 (Korn-Lichtenstein equations solved in the function space of bivariate polynomi-
als).

If a polynomial ( 15.29 )-( 15.32 )ofdegree n fulfills the Korn-Lichtenstein equations ( 15.27 ) with
respect to an ellipsoid-of-revolution and subject to the n − 1 constraints given by ( 15.52 )-( 15.64 ),

then the following mixed coecient relations hold.

n = 1 : (15.69)

y 10 = −r 0 x 01 , 01 = s 0 x 10 .

n = 2 : (15.70)

2 y 20 = −r 0 x 11 , 11 = − 2 r 0 x 02 − r 1 x 01 , 11 =2 s 0 x 20 , 2 y 02 = s 0 x 11 + s 1 x 10 .

n =3:

3 y 30 = − r 0 x 21 , 2 y 21 = − 2 r 0 x 12 − r 1 x 11 , 12 = − 3 r 0 x 03 − 2 r 1 x 02 − r 2 x 01 , (15.71)

y 21 =3 s 0 x 30 , 2 y 12 =2 s 0 x 23 +2 s 1 x 20 ,, 3 y 03 = s 0 x 12 + s 1 x 11 + s 2 x 10 .
n =4:

4 y 40 = − r 0 x 31 , 3 y 31 = − 2 r 0 x 22 − r 1 x 21 , 2 y 22 = − 3 r 0 x 13 − 2 r 1 x 12 − r 2 x 11 , (15.72)

y 13 = r 3 x 01 , 31 =4 s 0 x 40 , 2 y 22 =3 s 0 x 31 +3 s 1 x 30 ,

3 y 13 =2 s 0 x 22 +2 s 1 x 21 +2 s 2 x 20 , 4 y 04 = s 0 x 13 + s 1 x 12 + s 2 x 11 + s 3 x 10 .
In general

− 4 r 0 x 04 − 3 r 1 x 03 − 2 r 2 x 02 −

∞ n− 1 ∞ n− 1 i

( n − i ) y n−i,i l n−i− 1 b i = − ( i − j +1) r j x n− 1 −i,i−j +1 l n−i− 1 b i =

y l =

n =1 i =0 n =1 i =0 j =0

= − r ( b ) x b , (15.73)

∞ n− 1 ∞ n− 1 i

i ) y n−i− 1 ,i +1 l n−i− 1 b i = i ) s j x n−i,i−j l n−i− 1 b i =

y b = ( n − ( n −

n =1 i =0 n =1 i =0 j =0

= s ( b ) x l .

End of Corollary.

15-3 Constraints to the Korn-Lichtenstein Equations
(Gauss-Krueger/UTM Mappings)

The constraints to the Korn-Lichtenstein equations generating the Gauss-Krueger conformal
mapping or the UTM conformal mapping.

The equidistant mapping of a meridian of reference L 0 immediately establishes the proper
constraint to the Korn-Lichtenstein equations which leads to the standard Gauss-Krueger