376 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
The equations (16.51), which represent locally the oblique Mercator projection, reduce (i) to the
equations of the standard Mercator projection of E
2
A 1 ,A 2
for zero inclination, see Box 16.6, or (ii) to the
equations of the transverse Mercator projection of E
2
A 1 ,A 2
for ninety degrees inclination, see Box 16.7,
or (iii) to the equations of the oblique Mercator projection of S
2
r for zero relative eccentricity E = 0,
compare with Box 16.1 presented already before.
Box 16.6 (The equations of the standard Mercator projection of E
2
A 1 ,A 2 for zero inclination).
i = 0
⇒
E
= 0 , A
1 = A 1 , A
2 = A 1 ,
(16.73)
tan(L − Ω) = tan α
⇒
α = L − Ω ,
(16.74)
tan B = 0
⇒
b j = 0 ∀ j = 1, 2, . . . ,
(16.75)
s 1 (α) = A
1 = a , s 2 (α) = 0 , s 3 (α) = 0 , . . . ,
(16.76)
α 1 = 0 , α 2 = 0 , β 1 = s 1 = a , β 2 = 0
⇒
∆x = a∆l , ∆y = a∆q = aq 1 ∆b + aq 2 ∆b
2 + O 3 .
(16.77)
Box 16.7 (The equations of the transverse Mercator projection of E
2
A 1 ,A 2 for ninety degrees inclination).
i = π/2
⇒
E
= E , A
1 = A 1 , A
2 = A 1
p
1 − E 2 ,
(16.78)
tan(L − Ω) = 0
⇒
L = Ω ,
(16.79)
tan B = tan α/
√
1 − E 2 ,
(16.80)
s 1 (α) = A 1
√
1 − E 2 cos 2 α , s 2 (α) =
1
2
A 1 E
2 sin α cos α
√
1 − E 2 cos 2 α
,
(16.81)
l 1 = 0 , l 2 = 0 , . . . ,
(16.82)
b 1 =
dB
dα
(α 0 ) =
p
1 − E 2 /(1 − E
2 cos
2 α 0 ) =
cos
2 B 0
√
1 − E 2
[1 + (1 − E
2 ) tan
2 B 0 ] , b 2 = · · · . (16.83)