370 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
Theorem 16.5 (Universal oblique Mercator projection).
The boundary condition of the equidistantly mapped elliptic meta-equator E
1
A
1 ,A
2
,
∆x(meta-equator) = ∆s(∆α) , ∆y(meta-equator) = 0 ,
(16.45)
with respect to first power series s(α),
∆s(∆α) = s 1 ∆α + s 2 ∆α
2 + O s3 , ∆α := α − α 0 ,
(16.46)
the second power series B(α) and L(α),
∆b(∆α) = b 1 ∆α + b 2 ∆α
2 + O b3 ,
∆b := B − B 0 ,
(16.47)
∆l(∆α) = l 1 ∆α + l 2 ∆α
2 + O l3 ,
∆l := L − L 0 ,
(16.48)
∆b
2 (∆α) = b
2
1 ∆α
2 + O
2
b3 , ∆l
2 (∆α) = l
2
1 ∆α
2 + O
2
l3 ,
(16.49)
and the third power series q UMP (B),
∆q = q 1 ∆b + q 2 ∆b
2 + O q3 , ∆q := q UMP (B) − q UMP (B 0 ) ,
(16.50)
leads to the parameters of the second order universal oblique Mercator projection,
∆x =
= α 1 q 1 ∆b + β 1 ∆l + (α 1 q 2 + α 2 q
2
1 )∆b
2 + 2β 2 q 1 ∆b∆l − α 2 ∆l
2 + O x3 ,
∆y =
= β 1 q 1 ∆b − α 1 ∆l + (β 1 q 2 + β 2 q
2
1 )∆b
2
− 2α 2 q 1 ∆b∆l − β 2 ∆l
2 + O y3 ,
(16.51)
namely
α 1 =
q 1 b 1 s 1
q 2
1 b 2
1 + l 2
1
, β 1 =
l 1 s 1
q 2
1 b 2
1 + l 2
1
,
(16.52)
α 2 =
1
(q 2
1 b 2
1 + l 2
1 ) 3 ×
×
s 2 (q
2
1 b
2
1 − l
2
1 )(q
2
1 b
2
1 + l
2
1 ) + s 1 [(q 1 b 2 + q 2 b
2
1 )(3l
2
1 − q
2
1 b
2
1 )q 1 b 1 − l 1 l 2 (3q
2
1 b
2
1 − l
2
1 )]
,
(16.53)
β 2 =
1
(q 2
1 b 2
1 + l 2
1 ) 3 ×
×
2q 1 b 1 l 1 (q
2
1 b
2
1 + l
2
1 )s 2 + s 1 [(q 1 b 2 + q 2 b
2
1 )(−3q
2
1 b
2
1 + l
2
1 )l 1 + (−3l
2
1 + q
2
1 b
2
1 )q 1 b 1 l 1 ]
.
(16.54)
End of Theorem.