306 14 “Ellipsoid-of-revolution to cylinder”: polar aspect
To this end, we review the mapping equations and the principal stretches. They are easily computed
as follows.
x
y
=
⎡
⎣
A 1 Λ
A 1 (1−E
2 )
4E
ln
1+E sin Φ
1−E sin Φ
+
2E sin Φ
1−E 2 sin 2 Φ
⎤
⎦ ,
(14.29)
Λ 1 =
1 − E 2 sin
2 Φ
cos Φ
, Λ 2 =
cos Φ
1 − E 2 sin
2 Φ
.
(14.30)
14-24 Summary (cylindric mapping equations)
For the convenience of the reader, the central formulae that specify the mapping equations and the
principal stretches are summarized in the following Box 14.1.
Box 14.1 (Summary).
Type 1 (equidistant on the set of parallel circles):
x = A 1 Λ , y = f (Φ) ,
(14.31)
f (Φ) = A 1
„
E(π/2, E) − E
»
π
2
− arctan
„
A 2
A 1
tan Φ, E
«–«
,
(14.32)
Λ 1 =
p
1 − E 2 sin
2 Φ
cos Φ
, Λ 2 = 1 .
(14.33)
“Elliptic integral”.
Type 2 (normal conformal):
x = A 1 Λ , y = f (Φ) ,
(14.34)
f (Φ) = A 1 ln
"
tan
„
π
4
+
Φ
2
« „
1 − E sin Φ
1 + E sin Φ
« E/2
#
,
(14.35)
Λ 1 = Λ 2 =
p
1 − E 2 sin
2 Φ
cos Φ
.
(14.36)
Type 3 (normal equiareal):
x = A 1 Λ , y = f (Φ) ,
(14.37)
f (Φ) =
A 1 (1 − E
2 )
4E
»
ln
„
1 + E sin Φ
1 − E sin Φ
«
+
2E sin Φ
1 − E 2 sin
2 Φ
–
,
(14.38)
Λ 1 =
p
1 − E 2 sin
2 Φ
cos Φ
, Λ 2 =
cos Φ
p
1 − E 2 sin
2 Φ
.
(14.39)