Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 712

π
2 ,

B ( Q ) = 2 arctan exp Q − (D.60)

N→∞

arctan u = π

1
u +

1
3 u 3 −

1
5 u 5 +

1
2 r +1

1
u 2 r +1 =

1) r +1

2 −

( −

r =3

N→∞

= π ( − 1) r +1 exp[ − (2 r +1) Q ]

2 r +1

2 +

. (D.61)

r =0

We are thus led to the series representation of the boundary condition, for example, to ( D.62 )
and ( D.63 ), where we have eliminateed the coecients

{ A m ,B m } by the postulate q →∞ , { x, y }

finite.

mq )= r π

,

M→∞ N→∞

1) n +1 exp[

− (2 n +1) Q ]

2 n +1

x { q,p =0 } = x 0 + C m exp( −

2 −

2 ( − (D.62)

m =1 n =0

M→∞

y { q,p =0 } = y 0 + D m exp( − mq )=0 . (D.63)

m =1

Once we compare the coecients, we find ( D.64 ). Finally, we find the local representation of the
transverse Mercator projection in terms of isometric longitude and latitude p/q ,namely( D.65 ),
and in terms of longitude and latitude { l = L − L 0 ,B } ,namely( D.66 ).

x 0 = r π

2 ,y 0 =0 ,
C 2 n =0 ,C 2 n +1 = r 2(

1) n +1
2 n +1

−

,D m =0 , (D.64)

N→∞

x ( q,p )= r π

1
2 n +1 exp[

1) n +1

2 +2 r

( − − (2 n +1) q ]cos(2 n +1) p, (D.65)

n =0

N→∞

1
2 n +1 exp[

1) n

y ( q,p )=2 r ( − − (2 n +1) q ]sin(2 n +1) p,

n =0

N→∞

x ( B,l )= r π

1
2 n +1

cos(2 n +1) l
[tan( 4 −

1) n +1

2 +2 r

( −

2 )] 2 n +1 ,

(D.66)

B

n =0

N→∞

sin(2 n +1) l
[tan( 4 −

1
2 n +1

1) n y ( B,l )=2 r ( −

2 )] 2 n +1 .

B

n =0

End of Example.