284 10 “Sphere to cylinder”: polar aspect
Proof (distance preserving mapping).
Finally, we present the mapping equations of distance preserving type constrained to the postulate of
an equidistance mapping on two parallel circles. We have to specify the principal stretches as (10.40).
x
y
= R
Λ cos Φ 0
Φ
,
(10.39)
Λ 1 = cos Φ 0 / cos Φ , Λ 2 = 1 .
(10.40)
Starting from the above relations, we obtain
(Λ 1 − 1)
2 + (Λ 2 − 1)
2
2
=
1
2
(cos Φ 0 / cos Φ − 1)
2 ,
(10.41)
I A (equidistant) =
1
2
I A (conformal)
⇒
(10.42)
Φ = 85
◦ ,
Φ 0
= 61.72
◦ ,
I A = 0.3837 .
(10.43)
End of Proof.
In the following chapter, let us continue studying the mapping of the sphere to the cylinder, namely
let us study the transverse aspect.