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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
,β
1
,α
2
,β
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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