17-2 Special mapping equations 387
Fig. 17.8. Mapping the sphere to a cone. Polar aspect, conformal mapping, equidistant on the standard
parallel Φ = Φ 0 = 45
◦ .
17-222 Equidistance on two parallels (secant cone, J. H. Lambert 1772), compare with Fig. 17.9
The basic idea is to determine the cone constant n = sin Φ 0 from an equidistant mapping of two
standard parallel circles Φ = Φ 1 and Φ = Φ 2 . S tarting from (17.31), we immediately arrive at (17.32 ),
from which the integration constant c according to (17.33) is computed as a function of the unk nown
cone constant n. S ince c can also be determined via Λ 1 | Φ=Φ 2 = Λ 2 | Φ=Φ 2 : = 1, the equality (17.34 ) is
used to compute n according to (17.35 ).
Λ 1 = Λ 2 =
cn tan
π
4 −
Φ
2
n
R cos Φ
,
(17.31)
Λ 1 | Φ=Φ 1 = Λ 2
Φ=Φ 1
=
cn tan
π
4 −
Φ 1
2
n
R cos Φ 1
= 1 ,
(17.32 )
c =
R cos Φ 1
n tan
π
4 −
Φ 1
2
n ,
(17.33)
R cos Φ 1
n tan
π
4 −
Φ 1
2
n =
R cos Φ 2
n tan
π
4 −
Φ 2
2
n ,
(17.34 )
n =
ln cos Φ 1 − ln cos Φ 2
ln tan
π
4 −
Φ 1
2
− ln tan
π
4 −
Φ 2
2
.
(17.35 )
Fig. 17.8. Mapping the sphere to a cone. Polar aspect, conformal mapping, equidistant on the standard
parallel Φ = Φ 0 = 45
◦ .
17-222 Equidistance on two parallels (secant cone, J. H. Lambert 1772), compare with Fig. 17.9
The basic idea is to determine the cone constant n = sin Φ 0 from an equidistant mapping of two
standard parallel circles Φ = Φ 1 and Φ = Φ 2 . S tarting from (17.31), we immediately arrive at (17.32 ),
from which the integration constant c according to (17.33) is computed as a function of the unk nown
cone constant n. S ince c can also be determined via Λ 1 | Φ=Φ 2 = Λ 2 | Φ=Φ 2 : = 1, the equality (17.34 ) is
used to compute n according to (17.35 ).
Λ 1 = Λ 2 =
cn tan
π
4 −
Φ
2
n
R cos Φ
,
(17.31)
Λ 1 | Φ=Φ 1 = Λ 2
Φ=Φ 1
=
cn tan
π
4 −
Φ 1
2
n
R cos Φ 1
= 1 ,
(17.32 )
c =
R cos Φ 1
n tan
π
4 −
Φ 1
2
n ,
(17.33)
R cos Φ 1
n tan
π
4 −
Φ 1
2
n =
R cos Φ 2
n tan
π
4 −
Φ 2
2
n ,
(17.34 )
n =
ln cos Φ 1 − ln cos Φ 2
ln tan
π
4 −
Φ 1
2
− ln tan
π
4 −
Φ 2
2
.
(17.35 )