16-3 The equations of the oblique Mercator projection 371
The coefficients {q 1 , q 2 }, {s 1 , s 2 }, {b 1 , b 2 }, and {l 1 , l 2 } are collected in the following Boxes 16.2–16.5.
Box 16.2 (Isometric latitude q(b) as a function of latitude b).
Power series expansion ∆q =
P N
r=1 q r ∆b
r up to order N = 2
(higher-order terms are given by J. Engels, E. Grafarend (1995)):
q 1 :=
1 − E
2
cos B 0 (1 − E 2 sin
2 B 0 )
,
q 2 :=
sin B 0
2 cos 2 B 0 (1 − E 2 sin
2 B 0 ) 2 [1 + E
2 (1 − 3 sin
2 B 0 ) + E
4 (−2 + 3 sin
2 B 0 )] .
(16.55)
Box 16.3 (Arc length of the oblique meta-equator).
Power series expansion ∆s =
P N
r=1 s r ∆α
r up to order N = 2:
s 1 (α 0 ) := A 1
p
1 − E 2 cos 2 α 0 ,
s 2 (α 0 ) :=
1
2
A
1 E
2 sin α 0 cos α 0
p
1 − E 2 cos 2 α 0
.
(16.56)
Box 16.4 (Latitude B(α) as a function of meta-longitude α).
Power series expansion ∆b =
P N
r=1 b r ∆α
r up to order N = 2:
b 1 := −
A
2 A
1
2 (1 − E
2 ) sin i cos α 0
[A
1
2 (1 − E 2 ) 2 + E 2 A
2
2 sin
2 i sin
2 α 0 ]
[A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α 0 ]
−1/2 ,
2b 2 := −
A
2 A
1
2 (1 − E
2 ) sin i
[A
1
2 (1 − E 2 ) 2 + E 2 A
2
2 sin
2 i sin
2 α 0 ] 2 [A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α 0 ]
−3/2 ×
×
„
[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α 0 ][A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α 0 ] sin α 0 +
+2[A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α 0 ]E
2 A
2
2 sin
2 i sin α 0 cos
2 α 0 +
+[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α 0 ](A
2
2 cos
2 i − A
1
2 ) sin α 0 cos
2 α 0
«
.
(16.57)
Box 16.5 (Longitude L(α) as a function of meta-longitude α).
Power series expansion ∆l =
P N
r=1 l r ∆α
r up to order N = 2:
l 1 := +
A
1
2 A
2
2 cos i
A
1
2 cos 2 α 0 + A
2
2 cos 2 i sin
2 α 0
,
2l 2 := −2
A
1
2 A
2
2 cos i(A
2
2 cos
2 i − A
1
2 ) sin α 0 cos α 0
(A
1
2 cos 2 α 0 + A
2
2 cos 2 i sin
2 α 0 ) 2
.
(16.58)