386 17 “Sphere to cone”: polar aspect
17-22 Conformal mapping (Lambert projection)
The general mapping equations for this type of mappings are derived from the identity (17.22). The
mapping equations are obtained as (17.25). The left principal stretches are obtained as (17.26).
Λ 1 =
C 11
G 11
=
nf
R cos Φ
= Λ 2 =
C 22
G 22
=
f
R
(17.22)
⇓
f
f
=
n
cos Φ
⇒
df
f
= n
dΦ
cos Φ
(17.23)
⇓
ln f = n ln tan
π
4
−
Φ
2
+ ln c ,
(17.24)
α
r
=
nΛ
c tan
π
4 −
Φ
2
n
,
(17.25)
Λ 1 = Λ 2 =
cn tan
π
4 −
Φ
2
n
R cos Φ
.
(17.26)
17-221 Equidistance on the circle-of-contact, compare with Fig. 17.8
The constant n is defined using the parallel circle Φ = Φ 0 which shall be mapped equidistantly, i. e.
through the cone constant n = sin Φ 0 . It follows from (17.26) that (17.27) holds.
Λ 1 | Φ=Φ 0 = Λ 2 | Φ=Φ 0 =
cn tan
π
4 −
Φ 0
2
n
R cos Φ 0
= 1
⇔
c =
R cos Φ 0
n tan
π
4 −
Φ 0
2
n =
R cot Φ 0
tan
π
4 −
Φ 0
2
n .
(17.27)
The mapping equations for this kind of projection are therefore defined through (17.28) or (17.29).
The left principal stretches are provided by (17.30).
α
r
=
⎡
⎢
⎢
⎣
Λ sin Φ 0
R cot Φ 0
tan
π
4 −
Φ
2
tan
π
4 −
Φ 0
2
n
⎤
⎥
⎥
⎦ ,
(17.28)
x
y
= R cot Φ 0
tan
π
4 −
Φ
2
tan
π
4 −
Φ 0
2
n
cos(Λ sin Φ 0 )
sin(Λ sin Φ 0 )
,
(17.29)
Λ 1 = Λ 2 =
cos Φ 0
cos Φ
tan
π
4 −
Φ
2
tan
π
4 −
Φ 0
2
n
.
(17.30)