308 14 “Ellipsoid-of-revolution to cylinder”: polar aspect
Continuation of Box.
Left principal stretches:
Λ 1 =
p
c 11 /G 11 =
F (0)
F (Φ)
, Λ 2 =
p
c 22 /G 2 =
f
(Φ)
p
F 2 (Φ) + G 2 (Φ)
.
(14.47)
Structure of the coordinate lines:
(i): x = F (0)Λ .
(14.48)
(Straight line through the origin for Λ = const.)
(ii): y = f (Φ) .
(14.49)
(Straight line through the origin for Φ = const.)
14-31 Special normal cylindric mapping (equidistant: equator, set of parallel circles)
We start off by the postulate of an equidistant mapping on the set of parallel circles. This leads to
the following result.
Λ 2 = 1 ⇒
f
(Φ)
F 2 (Φ) + G 2 (Φ)
= 1 ⇔ df =
F 2 (Φ) + G 2 (Φ)dΦ ,
f (Φ) =
Φ
0
F 2 ( ˜
Φ) + G 2 ( ˜
Φ)d ˜
Φ + const. , f(0) = 0 ⇒ const. = 0 ,
(14.50)
x
y
=
⎡
⎣
F (0)Λ
Φ
0
F 2 ( ˜
Φ) + G 2 ( ˜
Φ)d ˜
Φ
⎤
⎦ , Λ 1 =
F (0)
F (Φ)
, Λ 2 = 1 .
(14.51)
14-32 Special normal conformal cylindric mapping (equidistant: equator)
Alternatively, let us start off by the postulate of a conformal mapping. Similar as before, this leads to
the following result.
Λ 1 = Λ 2 ⇒
F (0)
F (Φ)
=
f
(Φ)
F 2 (Φ) + G 2 (Φ)
⇔ df = F (0)
F 2 (Φ) + G 2 (Φ)
F (Φ)
dΦ ,
f (Φ) =
Φ
0
F (0)
F 2 ( ˜
Φ) + G 2 ( ˜
Φ)
F ( ˜
Φ)
d ˜
Φ + const. , f(0) = 0 ⇒ const. = 0 ,
(14.52)
x
y
= F (0)
⎡
⎣
Λ
Φ
0
√
F 2 ( ˜
Φ)+ G 2 ( ˜
Φ)
F ( ˜
Φ)
d ˜
Φ
⎤
⎦ , Λ 1 = Λ 2 =
F (0)
F (Φ)
.
(14.53)