554 E Geodesics
Box E.1 (The conformal coordinates as functions of spherical longitude and spherical latitude and the factor
of conformality).
(i) Universal Polar Stereographic Projection (UPS):
q
1 = 2R tan
„
π
4
−
Φ
2
«
cos Λ , q
2 = 2R tan
„
π
4
−
Φ
2
«
sin Λ ,
(E.41)
λ = cos
2
„
π
4
−
Φ
2
«
=
4R
2
4R 2 + (q 1 ) 2 + (q 2 ) 2 ,
(E.42)
ds
2 =
16R
4
(4R 2 + (q 1 ) 2 + (q 2 ) 2 ) 2
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.43)
(ii) Universal Mercator Projection (UM):
q
1 = R Λ , q
2 = R ln tan
„
π
4
+
Φ
2
«
= R artanh (sin Φ) ,
(E.44)
λ = cos Φ =
1
cosh q 2 /R
,
(E.45)
ds
2 =
1
cosh
2 q 2 /R
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.46)
(iii) Universal Transverse Mercator Projection (UTM):
q
1 = R arctan
tan Φ
cos(Λ − Λ 0 )
, q
2 = R artanh (− cos Φ sin(Λ − Λ 0 )) ,
(E.47)
λ = cos [arcsin(− cos Φ sin(Λ − Λ 0 ))] ,
(E.48)
cos Φ = cos
q
1
R
1 + tan
2 q
1
R
tanh
2 q
2
R
, sin(Λ − Λ 0 ) = − tanh
q
2
R
/ cos Φ ,
(E.49)
λ = cos
»
arcsin
„
tanh
q
2
R
«–
,
(E.50)
ds
2 = cos
2
»
arcsin
„
tanh
q
2
R
«– ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.51)
(iv) Universal Conic Projection (UC):
q
1 = r cos α , q
2 = r sin α , α = nΛ , r = c
„
tan
„
π
4
−
Φ
2
«« n
,
(E.52)
λ =
r cos Φ
cn
`
tan
` π
4
−
Φ
2
´´ n .
(E.53)