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1+
tanh
2
x/
1
tanh
2
x
=
x
1+1+
1
tanh
x
1+
1
2
+
1
2
+
1
−
−
2
2
−tanh
3
x
1
6
+
1
1
10
tanh
2
x
+
O
7
(
tanh x
)
−
(23.51)
6
H
2
onto a developed cylinder
C
1
Fig. 23.11.
Conformal mapping
.
Figures
23.11
,
23.12
,and
23.13
are a visualization of various hyperboloid map projections where
as a theme a
cardoid
onto a hyperboloid
2
has been mapped as an object.
This contribution is based on
Grafarend and Syffus
(
1997f
).
H
23-3 Mapping the Paraboloid: Projective Geometry of the Parabolic
Mirror with Boundary
2
,
is comparatively represented
in terms of charts of type
oblique orthogonal projection versus cylindric/conic,
namely of type
conformal, equiareal and equidistant.
Computer graphical examples are given.
The two-dimensional paraboloid
The
parabolic mirror,
geometrically described as the
paraboloid
P
2
P
of rotational symmetry as an algebraic manifold is
defined by
2
:=
3
X
2
+
Y
2
P
{
X
∈
R
|
−
Z
=0
}
for instance being represented in the
chart
{
parabolic longitude
U
, parabolic vertical
V }
∈{
R
2
|
0
<U<
2
π,
0
<V <
2
}
and the
boundary V
=2 by means of
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