206 5 “Sphere to tangential plane”: polar (normal) aspect
Box 5.22 (Wiechel’s pseudo-azimuthal projection “sphere to plane”).
Direct equations of Wiechel’s map (polar coordinates):
α = Λ +
∆
2
= Λ +
„
π
4
−
Φ
2
«
=: g(Λ, ∆) ,
r = 2R sin
∆
2
= 2R sin
„
π
4
−
Φ
2
«
=: f (∆) .
(5.147)
Direct equations of Wiechel’s map (Cartesian coordinates):
"
x
y
#
= 2R sin
∆
2
"
cos
`
Λ +
∆
2
´
sin
`
Λ +
∆
2
´
#
,
"
x
y
#
= f (∆)
"
cos g(Λ, ∆)
sin g(Λ, ∆)
#
.
(5.148)
Left Jacobi matrix:
J l =
"
D Λ x D ∆ x
D Λ y D ∆ y
# " −f sin g f
cos g − fg ∆ sin g
+f cos g f
sin g + fg ∆ cos g
#
, f
= R cos
∆
2
, g ∆ =
1
2
.
(5.149)
Left Cauchy–Green matrix:
G r = I 2 , C l = J
∗
l G r J l = J
∗
l J l =
"
f
2
1
2
f
2
1
2
f
2 f
2 + f
2 g
2
∆
#
=
"
4R
2 sin
2 ∆
2
2R
2 sin
2 ∆
2
2R
2 sin
2 ∆
2
R
2
#
.
(5.150)
Left principal stretches:
Λ
2
1,2 = Λ
2
± =
1
2
„
tr
ˆ
C l G
−1
l
˜ ±
q `
tr
ˆ
C l G
−1
l
˜´ 2 − 4det
ˆ
C l G
−1
l
˜
«
,
(5.151)
C l G
−1
l
=
2
6
6
6
4
1
cos 2 ∆
2
2 sin
2 ∆
2
1
2
1
cos 2 ∆
2
1
3
7
7
7
5
,
tr
ˆ
C l G
−1
l
˜
=
1 + cos
2 ∆
2
cos 2 ∆
2
, det
ˆ
C l G
−1
l
˜
= 1 ,
`
tr
ˆ
C l G
−1
l
˜´ 2 − 4det
ˆ
C l G
−1
l
˜
=
1
cos 4 ∆
2
" „
1 + cos
2 ∆
2
« 2
− 4 cos
4 ∆
2
#
,
(5.152)
Λ
2
1,2 = Λ
2
± =
1
2
1
cos 2 ∆
2
2
4 1 + cos
2 ∆
2
±
s „
1 + cos 2 ∆
2
« 2
− 4 cos 4 ∆
2
3
5
⇒
Λ 1 Λ 2 = 1 .
(5.153)