212 6 “Sphere to tangential plane”: transverse aspect
Lemma 6.1 (Transverse conformal mapping of the sphere to a tangential plane at the meta-North Pole
Λ 0 ∈ [0
◦ , 360
◦ ], Φ 0 = 0
◦ ).
x = 2R tan
π
4
−
B
2
cos A , y = 2R tan
π
4
−
B
2
sin A ,
tan A =
sin(Λ − Λ 0 )
− tan Φ
, sin B = cos Φ cos(Λ − Λ 0 ) ,
(6.10)
subject to the principal stretches
Λ 1 = Λ 2 =
1
cos 2
π
4 −
B
2
(6.11)
and the left Cauchy-Green eigenspace
left CG eigenspace =
E A
1
cos 2
π
4 −
B
2
, E B
1
cos 2
π
4 −
B
2
.
(6.12)
End of Lemma.
An idea of the appearance of the transverse conformal mapping to be considered here can be obtained
from Fig. 6.3.
Fig. 6.3. Mapping the sphere to a tangential plane: transverse aspect, conformal mapping. Point-of-contact:
meta-North Pole at Λ 0 = 90
◦ , Φ 0 = 0
◦ .