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α 1 Δl + β 2 ( Δq 2
Δl 2 )
= β 1 Δq
α 2 2 ΔqΔl +O y 3 .
We are left with the problem to determine the unknown coecients α 1 1 2 2 etc. by a properly
chosen boundary condition we outline as follows.
Definition 16.4 (Universal oblique Mercator projection).
2
A 1 ,A 2
A conformal mapping of the ellipsoid-of-revolution
E
is called Universal oblique M ercator pro-
E 2 ) / 1
1
A 1 ,A 2
for A 1 = A 1 and A 2 = A 1 (1
jection if its oblique elliptic meta-equator
E
E 2 cos 2 i
is mapped equidistantly as a straight line.
End of Definition.
Theorem 16.5 (Universal oblique Mercator projection).
The boundary condition of the equidistantly mapped elliptic
meta-equator
1
A 1 ,A 2
,
Δx (meta-equator) = Δs ( Δα ) , Δ (meta-equator) = 0 ,
E
(16.45)
with respect to first power series s ( α ) ,
Δs ( Δα )= s 1 Δα + s 2 Δα 2 +O s 3 , Δα := α
α 0 ,
(16.46)
the second power series B ( α )and L ( y ) ,
Δb ( Δα )= b 1 Δα + b 2 Δα 2 +O b 3 ,
(16.47)
Δb := B − B 0 ,
Δl ( Δα )= l 1 Δα + l 2 Δα 2 +O l 3 ,
(16.48)
Δl := L − L 0 ,
Δb 2 ( Δα )= b 1 Δα 2 +O b 3 , Δ 2 ( Δα )= l 1 Δα 2 +O i 3 ,
(16.49)
andthethirdpowerseries q UMP ( B ) ,
Δq = q 1 Δb + q 2 Δb 2 +O q 3 , Δq := q UMP ( B )
q UMP ( B 0 ) ,
(16.50)
leads to the parameters of the second order universal oblique
Mercator projection,
Δx =
= α 1 q 1 Δb + β 1 Δl +( α 1 q 2 + α 2 q 1 ) Δb 2 +2 β 2 q 1 ΔbΔl
α 2 Δl 2 + O x 3 ,
(16.51)
Δy =
α 1 Δl +( β 1 q 2 + β 2 q 1 ) Δb 2
β 2 Δl 2 +O y 3 ,
= β 1 q 1 Δb
2 α 2 q 1 ΔbΔl
namely
q 1 b 1 s 1
q 1 b 1 + l 1 , 1 =
l 1 s 1
q 1 b 1 + l 1 ,
α 1 =
(16.52)
1
α 2 =
( q 1 b 1 + l 1 ) 3 ×
(16.53)
× s 2 ( q 1 b 1
l 1 )( q 1 b 1 + l 1 )+ s 1 ( q 1 b 2 + q 2 b 1 )(3 l 1
q 1 b 1 ) q 1 b 1
 
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