302 14 “Ellipsoid-of-revolution to cylinder”: polar aspect
At this point, let us finally review the principal stretches and let us finally give the general structure
of the coordinate lines.
Λ 1 =
c 11 /G 11 =
1 − E 2 sin
2 Φ
cos Φ
, Λ 2 =
c 22 /G 22 =
f
(Φ)(1 − E
2 sin
2 Φ)
3/2
A 1 (1 − E 2 )
, (14.5)
x = A 1 Λ , y = f (Φ) .
(14.6 )
14-2 Special mapping equations
Special mapping equations of cylindric mappings: normal equidistant, normal conformal, and normal
equiareal mappings.
N ext, we present special normal mappings of type equidistant mapping, conformal mapping, and
equiareal mapping as second postulates.
14-21 Special normal cylindric mapping (equidistant: parallel circles,
conformal: equator)
As it is shown in the following chain of relations, let us transfer the postulate of an eq uidistant mapping
on the set of parallel circles.
Λ 2 = 1 ⇒
f
(Φ)
A 1 (1 − E 2 )
(1 − E
2 sin
2 Φ)
3/2 = 1 ⇔ df = A 1 (1 − E
2 )
dΦ
(1 − E 2 sin
2 Φ) 3/2 ,
f (Φ) = A 1 (1 − E
2 )
Φ
0
dΦ
(1 − E 2 sin
2 Φ ) 3/2 .
(14.7 )
A 1
Φ
Φ
∗
Z
A 2 ∆
π
2
+ Φ
P
P
∗
tangent
normal
spherical surface
Fig. 14.1. Vertical section of the ellipsoid-of-revolution.