Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 709

Superposition of base functions gives the setups ( D.42 )and( D.43 ). The d'Alembert-Euler equa-
tions (Cauchy-Riemann equations) x p = y q and x q =

− y p then are specified by ( D.44 )and( D.45 ).

M

x ( q,p )= [ I m cosh mq cos mp + J m cosh mq sin mp + (D.42)

m =1

+ K m sinh mq cos mp + L m sinh mq sin mp ]+ x 0 ,

M

[ I m cosh m q cos m p + J m cosh m q sin m p +

y ( q,p )= (D.43)

m =1

+ K m sinh m q cos m p + L m sinh m q sin m p ]+ y 0 ,

M

x q = [ mI m sinh mq cos mp + mJ m sinh mq sin mp + (D.44)

m =1

+ mK m cosh mq cos mp + mL m cosh mq sin mp ] ,

M

[ −m I m cosh m q sin m p + m J m cosh m q cos m p−

y p =

m =1

m K m sinh m q sin m p + m L m sinh m q cos m p ] − (D.45)

⇔
I m = L m ,J m = −K m ,K m = J m ,L m = −I m .

End of Proof.

r , transverse Mercator projection).

Example D.1 ( S

2
r
based on the fundamental solution ( D.15 ), ( D.16 ), ( D.28 ), and ( D.29 ) of the d'Alembert-Euler
equationsc (Cauchy-Riemann equations). Let us depart from the equidistant mapping of the L 0
meta-equator, namely the boundary condition

As an example, let us construct the transverse Mercator projection locally for the sphere S

x = x { q ( L = L 0 ,B ) ,p ( L = L 0 ,B ) } = rB,y = y { q ( L = L 0 ,B ) ,p ( L = L 0 ,B ) } =0 . (D.46)

There remains the task to express the boundary conditions in the function space ( D.15 )
and ( D.16 ). There are two ways in solving this problem.

First choice.
A Taylor series expansion of B ( Q ) around B 0 ( Q 0 )leads directly to
B = arcsin(tanh Q ) ,
B = B 0 + ΔB = B 0 + b ∀ b := ΔB ,
Q = Q 0 + ΔQ = Q 0 + q ∀ q := ΔQ
⇒

(D.47)

N→∞

b = d 1 q + d 2 q 2 + d r q r

r =3

∀