Optimal Map Projections by Variational Calculus: Harmonic Maps - Map Projections: Cartographic Information Systems - page 567

Detailed Examples 22.1 and 22.2 follow to illustrate homogeneous polynomial of degree 2 and
3 under the Laplace-Beltrami harmonicity condition and the recurrence relations fixed to initial
data ξ 20 =1 ,ξ 30 =1.

Example 22.1 (homogenous polynomial of degree 2: POLY ξ ( q,l ;1)= ξ 2 , 0 q 2 + ξ 1 , 1 ql + ξ 0 , 2 l 2 ).

1st derivatives:

D q POLY ( q,l ;2)=2 ξ 2 , 0 q + ξ 1 , 1 ,D l POLY ( q,l, ;2)=2 ξ 2 , 0 l + ξ 1 , 1 q

2nd derivatives:

D q POLY ( q,l ;2)=2 ξ 2 , 0 , D l POLY ( q,l, ;2)=2 ξ 0 , 2
ΔPOLY ξ ( q,l ;2)=0 ↔ ξ 2 , 0 + ξ 0 , 2 =0

End of Example.

Corollary ( POLY ξ ( q,l ;)).

Given a harmonic polynomial POLY ξ ( q,l ;2) of degree 2. If ξ 2 , 0 =1then ξ 0 , 2 = − 1. ξ 1 , 1 is left

undermined. If ξ 2 , 0 =1 ,ξ 0 , 2 = − 1 ,ξ 1 , 1 =2then

POLY ξ ( q,l ;2)= q 2 l 2 +2 ql −

End of Corollary.

Example 22.2 (homogenous polynomial of degree 3: POLY ξ ( q,l ;3)= ξ 3 , 0 q 3 + ξ 2 , 1 q 2 l + ξ 1 , 2 ql 2 +
ξ 0 , 3 l 3 ).

1st derivatives

D q POLY ξ ( q.l ;3)=3 ξ 3 , 0 q 2 +2 ξ 2 , 1 ql + ξ 1 , 2 l 2
D l POLY ξ ( q,l, ;3)= ξ 2 , 1 q 2 +2 ξ 1 , 2 ql +3 ξ 0 , 3 l 2

2nd derivatives

D q POLY ξ ( q,l ;3)=3 × 2 ξ 3 , 0 q +2 ξ 2 , 1 l
D l POLY ξ ( q,l ;3)=2 ξ 1 , 2 q +3 × 2 ξ 0 , 3 l

ΔPOLY ξ ( q.l ;3) = 0 ↔ 3 × 2 ξ 3 , 0 q +2 ξ 1 , 2 q +2 ξ 2 , 1 l +3 × 2 ξ 0 , 3 l =0

6 ξ 3 , 0 +2 ξ 1 , 2 =0
2 ξ 2 , 1 +6 ξ 0 , 3 =0 ↔

ξ 1 , 2 =

3 ξ 3 , 0
ξ 2 , 1 = − 3 ξ 0 , 3

−

↔

End of Example.