Optimal Map Projections by Variational Calculus: Harmonic Maps - Map Projections: Cartographic Information Systems - page 570

Fig. 22.2. Recurrence relation of coecients, bivariate harmonic polynomial

1) k/ 2 r ξ r−k,k =( − (22.72)

k

k odd

1) ( k− 1) / 2 r ξ r−k,k =( − (22.73)

k

End of Corollary.

Table 22.8 Homogeneous bivariate harmonic polynomials of even and odd type

POLY ξ ( q,l ; r )= POLY ξ ( EVEN : q,l )+ POLY ξ ( ODD : q,l ) (22.74)

POLY ξ ( EVEN : q,l )=

r ( r − 1)

q r− 2 l 2 + r ( r

− 1) ( r − 2) ( r − 3) r ( r − 1) ( r − 2) ( r − 3) ( r − 4)( r − 5)

q r q r− 4 l 4 −

−

2 4 ∗ 3 ∗ 2 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2

1) k/ 2 r
k

q r−k l k

q r− 6 l 6 + r
k =8

( −

r ( r − 1)( r − 2)

q r− 3 l 3 + r ( r

− 1)( r − 2)( r − 3)( r − 4)

POLY ξ ( ODD : q,l )= rq r− 1 l −

3 ∗ 2 5 ∗ 4 ∗ 3 ∗ 2

1) k− 1 / 2 r
k

q r−k l k

q r− 5 l 5 ++

r
k =7

( − (22.75)

“ POLY η ( q,l ; r ) analogously

Table 22.9 Homogenous bivariate harmonic polynomials

( − 1) s r
2 s

q r− 2 s l 2 s + [ r +1

( − 1) s +1 r
2 s

q r− (2 s− 1) l 2 s− 1

POLY ξ ( q,l ; r )= [ 2 ]
s =0

]

2

(22.76)

− 1 s =1

1) s r
2 s

q r− 2 s l 2 s + [ r +1

1) s +1 r
2 s

q r− (2 s− 1) l 2 s− 1

POLY η ( q,l ; r )= [ 2 ]
s

]

2

( − ( − (22.77)

− 1

=0 s =1