Geodetic Mapping - Map Projections: Cartographic Information Systems - page 488

As a result, we compute and obtain the series (the coecients [ μν ]are

collected in Box 20.5 and have to be computed at the point B 0 )

{ u =0 ,v = y }

B 0 = [01] y + [02] y 2 + [03] y 3 + [04] y 4 +

B F − ··· (a polynomial in y ) , (20.113)

L F − L 0 =0 , α F0 − α 0F =0 ,

(20.114)

B F = B 0 +( B F − B 0 ) , L F = L 0 , α F0 = α 0F = π/ 2 .

Second step: determine L, B, γ, starting point
P F ,s = x = y c ,y = x c =0 ,α FP = α F0 +3 π/ 2 = 0 given.
As a result, we compute

, which leads to the series (the coecients [ μν ]are
collected in Box 20.5 and have to be computed at the point B F )

{ u = x, v =0 }

B − B F = [20] x 2 + [40] x 4 + ···

(an even polynomial in x ) ,

L F = [10] x + [30] x 3 + [50] x 5 + L − ··· (an odd polynomial in x ) , (20.115)

α PF = γ = [10] α x + [30] α x 3 +

···± π (an odd polynomial in x ) ,

B = B F +( B − B F ) ,

(20.116)

L = L F +( L − L F ) .

An alternative procedure is the following one-step method: set up a Taylor series for the
coecients [ μν ]in( 20.115 ), which have to be computed at the point B F , with B 0 as the
point of expansion, i.e.

0

∂B 2 0

∂ 2 [ μν ]

| 0 + ∂ [ μν ]
∂B

B 0 )+ 1
2!

B 0 ) 2 + hit([ μν ]) ,

[ μν ] | F =[ μν ] ( B F − ( B F − (20.117)

0 ( B F − B 0 )+ 1 0 ( B F − B 0 ) 2

[ μν ] α | F =[ μν ] α | 0 + ∂ [ μν ] α

∂ 2 [ μν ] α
∂B 2

∂B

2!

+hit([ μν ] α ) . (20.118)

B 0 ) 2 etc. are computed from the first step, for instance,

The terms B F − B 0 , ( B F −

0 y 2 + [03] 0 y 3 + [04] 0 y 4 +

B F − B 0 = [01] | 0 y + [02] | | | ··· ,

(20.119)

B 0 ) 2 = ([01] 0 y 2 + [03] 0 y 3 + [04] 0 y 4 + ) 2 etc .,

( B F − | 0 y + [02] | | | ···

leading to