The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 677

B
The Inverse of a Multivariate Homogeneous Polynomial

Univariate, bivariate, and multivariate polynomials and their inversion formulae. Cayley
multiplication and Kronecker-Zehfuss product. Triangular matrix.

For inversion problems of map projections like the computation of conformal coordinates of type
Gauss-Krueger or Universal Transverse Mercator Projection (UTM), or alternative coordinates
of type Riemann or Soldner-Fermi , we may take advantage of the inversion of (i) an univariate
homogeneous polynomial outlined in Sect. B-1 , (ii) a bivariate homogeneous polynomial outlined
in Sect. B-2 , or (iii) a trivariate , in general, multivariate homogeneous polynomial of degree n ,
which is discussed in Sect. B-3 .

Note that on the basis of an algorithm that is outlined
in Koenig and Weise ( 1951a , pp. 465-466, 501-511),
H. Glasmacher and K. Krack (1984, degree 6) as well
as Joos and Joerg ( 1991 , degree 5) have developed sym-
bolic computer manipulation software for the inversion of a
bivariate homogeneous polynomial.

Furthermore, solutions for the inversion of a univariate homogeneous polynomial are already
tabulated in Abramowitz and Stegun ( 1965 , p. 16, degree 7). However, we follow here Grafarend
( 1996 ), where the inversion of a general multivariate homogeneous polynomial of degree n suited
for symbolic computer manipulation is presented. For the mathematical foundation of the GKS
algorithm, we refer to Bass et al. ( 1982 ).

B-1 Inversion of a Univariate Homogeneous Polynomial of Degree n

Assume the univariate homogeneous polynomial of degree n ,namely y ( x )ofBox B.1 ,tobcgiven
and find the inverse univariate homogeneous polynomial of degree n ,namely x ( y ), i.e. from the
set of coecients

{ a 11 ,a 12 , ..., a 1 n− 1 ,a 1 n } , by the algorithm that is outlined in Box B.1 , find

the set of coecients { b 11 ,b 12 , ...,b 1 n− 1 ,b 1 n } .