558 E Geodesics
Box E.4 (The conformal coordinates as functions of ellipsoidal longitude and ellipsoidal latitude and the
factor of conformality).
(i) Universal Polar Stereographic Projection (UPS):
q
1 = f (Φ) cos Λ , q
2 = f (Φ) sin Λ ,
(E.68)
f (Φ) :=
2A
√
1 − e 2
„
1 − e
1 + e
« e
2
„
1 + e sin Φ
1 − e sin Φ
« e
2
tan
„
π
4
−
Φ
2
«
,
(E.69)
λ =
A cos Φ
p
1 − e 2 sin
2 Φ
1
f (Φ)
=
A cos f
−1
“ p
(q 1 ) 2 + (q 2 ) 2
”
r
1 − e 2 sin
2 f −1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1
p
(q 1 ) 2 + (q 2 ) 2
,
(E.70)
ds
2 =
A
2 cos
2 f
−1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1 − e 2 sin
2 f −1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1
(q 1 ) 2 + (q 2 ) 2
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.71)
(λ(q
1 , q
2 ) from series inversion of q
1 (Λ, Φ) and q
2 (Λ, Φ) leading to Λ(q
1 , q
2 ) and Φ(q
1 , q
2 ).
See J. P. Snyder (1987), pp. 162.)
(ii) Universal Mercator Projection (UM):
q
1 = A Λ , q
2 = A f(Φ) ,
(E.72)
f (Φ) := ln tan
„
π
4
+
Φ
2
« „
1 − e sin Φ
1 + e sin Φ
« e
2
,
(E.73)
λ =
p
1 − e 2 sin
2 Φ
cos Φ
=
p
1 − e 2 sin
2 f −1 (q 2 /A)
cos f −1 (q 2 /A)
,
(E.74)
ds
2 =
1 − e
2 sin
2 f
−1 (q
2 /A)
cos 2 f −1 (q 2 /A)
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.75)
(λ(q
2 ) from series inversion of q
2 (Φ) leading to Φ(q
2 ).
See J. P. Snyder (1987), pp. 45.)
(iii) Universal Transverse Mercator Projection (UTM):
q
1 = ρ[a 0 + a 10 b + a 20 b
2 + a 02 l
2 + a 30 b
3 + a 12 bl
2 + a 40 b
4 + a 22 b
2 l
2 + a 04 l
4 + a 50 b
5 +
+a 32 b
3 l
2 + a 14 bl
4 + O 1 (b
6 , l
6 )] (northern) ,
q
2 = ρ[a 01 l + a 11 bl + a 21 b
2 l + a 03 l
3 + a 31 b
3 l + a 13 bl
3 + a 41 b
4 l+
+a 23 b
2 l
3 + a 05 l
5 + O 2 (b
6 , l
6 )] (eastern) .
(E.76)
(Valid ∀ b := Φ − Φ 0 , l := Λ − Λ 0 , ρ := 0.999 578 (dilatation factor).
Coefficients see E. Grafarend (1994), Table 3.3.)
Box E.4 (The conformal coordinates as functions of ellipsoidal longitude and ellipsoidal latitude and the
factor of conformality).
(i) Universal Polar Stereographic Projection (UPS):
q
1 = f (Φ) cos Λ , q
2 = f (Φ) sin Λ ,
(E.68)
f (Φ) :=
2A
√
1 − e 2
„
1 − e
1 + e
« e
2
„
1 + e sin Φ
1 − e sin Φ
« e
2
tan
„
π
4
−
Φ
2
«
,
(E.69)
λ =
A cos Φ
p
1 − e 2 sin
2 Φ
1
f (Φ)
=
A cos f
−1
“ p
(q 1 ) 2 + (q 2 ) 2
”
r
1 − e 2 sin
2 f −1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1
p
(q 1 ) 2 + (q 2 ) 2
,
(E.70)
ds
2 =
A
2 cos
2 f
−1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1 − e 2 sin
2 f −1
“ p
(q 1 ) 2 + (q 2 ) 2
”
1
(q 1 ) 2 + (q 2 ) 2
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.71)
(λ(q
1 , q
2 ) from series inversion of q
1 (Λ, Φ) and q
2 (Λ, Φ) leading to Λ(q
1 , q
2 ) and Φ(q
1 , q
2 ).
See J. P. Snyder (1987), pp. 162.)
(ii) Universal Mercator Projection (UM):
q
1 = A Λ , q
2 = A f(Φ) ,
(E.72)
f (Φ) := ln tan
„
π
4
+
Φ
2
« „
1 − e sin Φ
1 + e sin Φ
« e
2
,
(E.73)
λ =
p
1 − e 2 sin
2 Φ
cos Φ
=
p
1 − e 2 sin
2 f −1 (q 2 /A)
cos f −1 (q 2 /A)
,
(E.74)
ds
2 =
1 − e
2 sin
2 f
−1 (q
2 /A)
cos 2 f −1 (q 2 /A)
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.75)
(λ(q
2 ) from series inversion of q
2 (Φ) leading to Φ(q
2 ).
See J. P. Snyder (1987), pp. 45.)
(iii) Universal Transverse Mercator Projection (UTM):
q
1 = ρ[a 0 + a 10 b + a 20 b
2 + a 02 l
2 + a 30 b
3 + a 12 bl
2 + a 40 b
4 + a 22 b
2 l
2 + a 04 l
4 + a 50 b
5 +
+a 32 b
3 l
2 + a 14 bl
4 + O 1 (b
6 , l
6 )] (northern) ,
q
2 = ρ[a 01 l + a 11 bl + a 21 b
2 l + a 03 l
3 + a 31 b
3 l + a 13 bl
3 + a 41 b
4 l+
+a 23 b
2 l
3 + a 05 l
5 + O 2 (b
6 , l
6 )] (eastern) .
(E.76)
(Valid ∀ b := Φ − Φ 0 , l := Λ − Λ 0 , ρ := 0.999 578 (dilatation factor).
Coefficients see E. Grafarend (1994), Table 3.3.)