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1
(
q
1
b
1
+
l
1
)
3
×
α
2
=
(16.72)
×
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
−l
1
l
2
(3
q
1
b
1
− l
1
)
,
1
(
q
1
b
1
+
l
1
)
3
×
β
2
=
×
2
q
1
b
1
l
1
(
q
1
b
1
+
l
1
)
s
2
+
s
1
(
q
1
b
2
+
q
2
b
1
)(
3
q
1
b
1
+
l
1
)
l
1
−
3
l
1
+
q
1
b
1
)
q
1
b
1
l
1
.
+(
−
End of Proof.
Equation (
16.51
), which represent locally the oblique Mercator projection, reduce (i) to the
equations of the standard Mercator projection of
2
A
1
,A
2
E
for zero inclination, see Box
16.6
, or (ii)
2
A
1
,A
2
for 90
◦
inclination, see Box
16.7
,
to the equations of the transverse Mercator projection of
E
2
or (iii) to the equations of the oblique Mercator projection of
S
r
for zero relative eccentricity
E
= 0, compare with Box
16.1
presented already before.
2
A
1
,A
2
Box 16.6 (The equations of the standard Mercator projection of
E
for zero inclination).
i
=0
⇒
(16.73)
E
=0
, A
1
=
A
1
, A
2
=
A
1
,
tan(
L
−
Ω
)=tan
α
⇒
(16.74)
α
=
L
Ω,
tan
B
=0
⇒
−
(16.75)
j
=1
,
2
,...,
s
1
(
α
)=
A
1
=
a, s
2
(
α
)=0
,s
3
(
α
)=0
,....,
b
j
=0
∀
(16.76)
α
1
=0
,α
2
=0
,β
1
=
s
1
=
a, β
2
=0
⇒
(16.77)
Δx
=
aΔl, Δy
=
aΔq
=
αq
1
Δb
+
αq
2
Δb
2
+O
3
.
2
A
1
,A
2
for 90
◦
inclination).
Box 16.7 (The equations of the transverse Mercator projection of
E
i
=
π/
2
⇒
(16.78)
A
2
=
A
1
√
1
E
=
E, A
1
=
A
1
,
−
E
2
,
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