Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 714

a cos B
120(1 − e 2 ) 3 1 − e 2 sin 2 B ×

−

a 5 =

5+28sin 2 B 24 sin 4 B + e 2 (1 + 16 sin 2 B 86 sin 4 B +72sin 6 B )+

× [ − − −

+ e 4 sin 4 B ( 26 + 100 sin 2 B 77 sin 4 B )+ e 6 sin 6 B (12 39 sin 2 B +28sin 4 B )] . − − −

Solution of the boundary value problem.
Let us consider the boundary value problem for the d'Alembert-Euler equations (Cauchy-
Riemann equations) subject to the integrability conditions of harmonicity type, namely ( D.72 )
and ( D.73 ), in the function space of polynomial type ( D.74 )and( D.75 ). Once we compare the
boundary conditions in the base {q r− 2 s p 2 s ,q r− 2 s +1 p 2 s− 1

} ,weareledto( D.76 ).

x p = y q ,x q = −y p ,
Δx = x pp + x qq =0 ,Δy = y pp + y qq =0 . (D.72)

x (0 ,q )= ∞ a r q r ,y (0 ,q )=0 ,

(D.73)

r =0

x ( p, q )= α 0 + α 1 q + β 1 p + α 2 ( q 2 p 2 )+2 β 2 qp +

−

( − 1) s r

q r− 2 s p 2 s +

( − 1) s +1 r
2 s

q r− 2 s +1 p 2 s− 1 , [ r/ 2] [( r +1) / 2]

N N

+ α r β r (D.74)

2 s

− 1

r =3 s =0 r =3 s =1

β 2 ( q 2 p 2 )+2 α 2 qp y ( p, q )= β 0 − β 1 q + α 1 p − − −

1) s r

q r− 2 s p 2 s +

1) s +1 r

q r− 2 s +1 p 2 s− 1 ,

[ r/ 2] [( r +1) / 2]

N N

− β r ( − α r ( − (D.75)

2 s 2 s − 1

r =3 s =0 r =3 s =1

α r = a r ( B 0 ) ,β r =0 ∀ r =0 ,...,∞. (D.76)

End of Example.

a,b , universal transverse Mercator projection modulo unknown dilatation param-
eter, c:c:cha-cha-cha).

Corollary D.5 ( E

The solution of the boundary value problem, where the L 0 meta-equator is modulo a dilatation
parameter equidistantly mapped, is given by ( D.77 ), ( D.78 ), and the a r of Table D.1 .

⎡

q r− 2 s p 2 s ⎤

( − 1) s r

[ r/ 2]

N

⎣ a 0 + a 1 q + a 2 ( q 2 ⎦ ,

− p 2 )+

x ( p, q )= a r

(D.77)

2 s

r =3 s =0

⎡

q r− 2 s p 2 s− 1 ⎤

1) s +1 r

[( r +1) / 2]

N

⎣ a 1 p +2 a 2 qp + ⎦ .

y ( p, q )= a r ( − (D.78)

2 s − 1

r =3 s =1