5-2 Special mapping equations 169
Box 5.4 (Conformal mapping of the sphere to the tangential plane at the North Pole).
Postulate of a conformeomorphism:
Λ 1 = Λ 2 ,
f (∆)
R sin ∆
=
f
(∆)
R
⇒
df
f
=
d∆
sin ∆
.
(5.26)
Integration of the characteristic differential equation of a conformal mapping S
2
R → T N S
2
R :
Z dx
sin x
= ln
˛
˛
˛tan
x
2
˛
˛
˛ ,
Z dy
y
= ln y ,
Z df
f
=
Z d∆
sin ∆
⇔ ln f = ln
˛
˛
˛
˛ tan
∆
2
˛
˛
˛
˛ + ln c , f(∆) = c tan
∆
2
∀ ∆ ∈ ]0, π[ .
(5.27)
Parameterized conformal mapping:
"
x
y
#
= 2R tan
∆
2
"
cos Λ
sin Λ
#
= 2R tan
„
π
4
−
Φ
2
« "
cos Λ
sin Λ
#
.
(5.28)
Left principal stretches:
Λ 1 = Λ 2 =
1
cos 2 ∆
2
=
1
cos 2
` π
4
−
Φ
2
´ .
(5.29)
Left eigenvectors:
C 1 Λ 1 = E Λ
1
cos 2
` π
4
−
Φ
2
´ (“Easting”) , C 2 Λ 2 = E Φ
1
cos 2
` π
4
−
Φ
2
´ (“Northing”) .
(5.30)
Left angular shear:
P
l = Ψ l − Ψ r = 0 , Ω l = 0 .
(5.31)
Parameterized inverse mapping:
tan Λ =
y
x
, tan
∆
2
=
1
2R
p
x 2 + y 2 .
(5.32)
Box 5.5 (Distortion analysis at the North Pole).
Postulate of an isometry at the North Pole:
lim
∆→0
Λ 2 (∆) = 1 .
(5.33)
Eigenspace analysis of the matrix pair {C l , G l }:
Λ 2 =
f
(∆)
R
, f
(∆) =
df
d∆
=
c
2
1
cos 2 ∆
2
⇒ Λ 2 =
c
2R
1
cos 2 ∆
2
,
Λ 2 =
c
2R
1
cos 2 ∆
2
, lim
∆→0
Λ 2 (∆) = 1⇒c = 2R ,
df
d∆
=
R
cos 2 ∆
2
, r = f (∆) = 2R tan
∆
2
.
(5.34)