Geography Reference
In-Depth Information
Box 5.13 (Data for the line-of-sight and the line-of-contact, critical spherical latitude, center
of perspective under the South Pole).
Tangential plane at the North Pole
Tangential plane at the South Pole
sin
|Φ
r
|
=
R
R
R
+
H
sin
|Φ
r
|
=
R
R
R
+
H
,
D
=
versus
D
=
r
max
R
+
D
=
r
max
2
R
+
H
=
r
max
H
tan
|
Φ
r
|
=
versus
tan
|
Φ
r
|
,
(5.76)
r
max
=(2
R
+
H
)tan
|
Φ
r
|
versus
r
max
=
H
tan
|
Φ
r
|
;
sin
x
1
−
sin
2
x
R
R
+
H
1
R
(2
R
+
H
)
H
;
tan
x
=
,
tan
|
Φ
r
|
=
1
=
(5.77)
R
(
R
+
H
)
2
−
r
max
=
R
1+2
R
H
R
1+2
H
versus
r
max
=
.
(5.78)
Let us compute the maximal extension of such a normal cent
ral p
erspective. According to the
identities of Box
5.1
3
, the
maximal extension
r
max
is either
R
√
1+
x
for a projection plane at the
North Pole or
R/
√
1+
x
for a projection plane at the South Pole and
x
:= 2
R/H
. Figure
5.15
and Table
5.1
outline those functions in the domain 0
≤
x
≤
5.
Example 5.1 (Numerical example I).
A first numerical example is
R/H
=3
/
2and
x
= 3, such that
√
1+
x
=2,1
/
√
1+
x
=1
/
2,
r
max
(North) = 2
R
,and
r
max
(South) =
R/
2.
End of Example.
Example 5.2 (Numerical example II).
A second numerical example is
R/H
=40and
x
= 80, such that
√
1+
x
=9
,
1
/
√
1+
x
=1
/
9,
r
max
(North) = 9
R
,and
r
max
(South) =
R/
9.
End of Example.
Obviously, by means of a normal central perspective from a southern perspective center to a
projection plane at the North Pole, we can cover more points than on the northern hemisphere.
In contrast, a normal central perspective from a southern perspective center to a projection plane
at the South Pole, we can cover only few points of the southern hemisphere.
Such a discussion motivates the construction of a
minimal atlas
from the setup of a normal
central perspective as follows. Consider the two charts (i) central perspective projection from a
southern perspective center to a projection plane at the North Pole and (ii) central perspective
projection from a northern perspective center to a projection plane at the South Pole. The union
of the two charts covers the sphere
2
S
R
completely. The two charts constitute a
minimal atlas
.
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