Optimal Map Projections by Variational Calculus: Harmonic Maps - Map Projections: Cartographic Information Systems - page 566

fulfill the harmonicity condition Eq.( 22.58 ). Here we have ordered the Laplace-Beltrami condition
of harmonicity into even POLY ξ ( EVEN : q,l )and odd POLY ξ ( ODD : q,l ).

Table 22.4 Taylor polynomials ansatz: solving the vector-valued Laplace-Beltrami equation on an ellipsoid of
revolution

“ansatz”

homogeneous polynomial of degree r

L Q 0

x = ξ ( q,l ) ,y = η ( q,l )

− L 0 =: l, q := Q −

Δ ξ ( q,l )
η ( q,l )

=0 ∂ 2

∂q 2

ξ ( q,l )
η ( q,l )

= 0

+ ∂ 2
∂l 2

↔ (22.53)

ξ ( q,l )= POLY ξ ( q,l,r )= r ξ hk q h l k = r

k ξ r−k,k q r−k l k

(22.54)

h,k =0
h + k = r

η ( q,l )= POLY η ( q,l,r )= r η hk q h l k = r

k =0

η r−k,k q r−k l k (22.55)

h,k =0
h + k = r

POLY ξ ( q,l,r )= ξ r, 0 q r + ξ r− 1 , 1 q r− 1 l + ξ r− 2 , 2 q r− 2 l 2 + ξ r− 3 , 3 q r− 3 l 3 + ξ r− 4 , 4 q r− 4 l 4 +
ξ r− 5 , 5 q r− 5 l 5 + ... + ξ 5 ,r− 5 q 5 l r− 5 + ξ 4 ,r− 4 q 4 l r− 4 + ξ 3 ,r− 3 q 3 l r− 3 + ξ 2 ,r− 2 q 2 l r− 2 +
ξ 1 ,r− 1 q 1 l r− 1 + ξ 0 ,r l r

(22.56)

POLY η ( q,l,r )= η r, 0 q r + η r− 1 , 1 q r− 1 l + η r− 2 , 2 q r− 2 l 2 + η r− 3 , 3 q r− 3 l 3 + η r− 4 , 4 q r− 4 l 4 +
η r− 5 , 5 q r− 5 l 5 + ... + η 5 ,r− 5 q 5 l r− 5 + η 4 ,r− 4 q 4 l r− 4 + η 3 ,r− 3 q 3 l r− 3 + η 2 ,r− 2 q 2 l r− 2 +
η 1 ,r− 1 q 1 l r− 1 + η 0 ,r l r

(22.57)

“decomposition in even and odd bivariate harmonic polynomials”

Δξ ( q,l )=0 ,Δη ( q,l )=0 ↔ Δ POLY ξ ( q,l,r )=0 ,Δ POLY η ( q,l,r )=0

1) q r− 2 + ξ r− 1 , 1 ( r − 2) q r− 3 l + ξ r− 2 , 2 ( r −

Δ POLY ξ ( q,l,r )= ξ r, 0 ( r − 1) ( r − 2) ( r − 3)

q r− 4 l 2 + ξ r− 3 , 3 ( r − 3)( r − 4) q r− 5 l 3 + ξ r− 4 , 4 ( r − 4)

5) q r− 6 l 4 + ... + ξ r− 2 , 2 q r− 2 + ξ r− 3 , 3 3 2 q r− 3 l + ξ r− 4 , 4 4 3 q r− 4 l 2 + ( r − ∗ ∗

4 q r− 5 l 3 + ... + ξ 4 ,r− 4 4 3 q 2 l r− 4 + ξ 5 ,r− 5 5 4 q 3 l r− 5 + ξ 2 ,r− 2 2 l r− 2 +

ξ r− 5 , 5 5 ∗ ∗ ∗

2 ql r− 3 + ξ 2 ,r− 2 ( r 3) q 2 l r− 4 + ξ 3 ,r− 3 ( r 4) q 3 l r− 5 +

ξ 3 ,r− 3 3 ∗ − 2)( r − − 3)( r −

2) ql r− 3
= Δ POLY ξ ( EVEN : q,l )+ Δ POLY ξ ( ODD : q,l ) = 0

1) l r− 2 + ξ 1 ,r− 1 ( r

ξ 0 ,r r ( r − − 1)( r −

(22.58)

Δ POLY η ( q,l,r )analogously

Table 22.5 is a collection of even bivariate harmonic polynomials highlighted by a corollary
for special recurrence relations of Eqs.( 22.59 )-( 22.62 ). Here we give an illustration by Fig. 22.2
and the examples of basis elements q r− 4 l 2 ,q r− 6 l 4 ,...,q 2 l r− 4 ,l r− 2

. Similarly Table 22.6 collects
the odd bivariate harmonic polynomials leading to a corollary for special recurrence relations of
Eqs.( 22.63 )-( 22.66 ). The examples contain the basis elements

{ }

q r− 3 l,q r− 5 l 3 ,...,q 3 l r− 5 ,ql r− 3 { } .