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 r | = R
R
R + H
sin 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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