Geography Reference
In-Depth Information
P 2 onto a developed cylinder
C 0
Fig. 23.21. Equidistant mapping of
23-4 Mapping the Onion Surface: Projected Geometrically
onto a Church Tower
2 , is comparatively represented
in terms of charts of type oblique orthogonal projection versus cylindric, namely of type conformal,
equiareal and equidistant. Computer graphical examples are given.
As a surface of revolution the church tower of onion shape may be defined by
The church power, geometrically described as the onion surface
Z
a 2 Z 2 / ( n +1) 1
Z 2 / ( n +1) =0 ,a
2 :=
3
X 2 + Y 2
+ ,n
Z
{
X
R
|
R
N
}
,
3
for instance represented in the chart
{
onion longitude U, onion latitude, V
}∈{ R
|
0 <U<
2 π, 0 <V <π/ 2
}
by means of
X = a ( cosV ) n sinVcosU = g ( V ) cosU
Y = a ( cosV ) n sinVsinU = g ( V ) sinU
(23.95)
( cosV ) n cosV =
( cosU ) n +1 = h ( V )
Z =
Figures 23.22 , 23.23 , 23.24 ,and 23.25 are a computer gra phic illust ration of the church tower
of onion shape , firstly by means of the “ onion function X 2 + Y 2 = aZ n/ ( n +1) 1 − Z n/ ( n +1)
and secondly by means of an oblique orthogonal projection which is generated in the following
way: A rotation around the 1-axis, 2-axis and 3-axis parameterized by means of Cardan angles
{
α
[0 , 2 π ]
[0 ]
[0 , 2 π ]
}
transforms ( X , Y , Z )into( X Y Z ), namely
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