5-2 Special mapping equations 187
Box 5.14 (Data for the line-of-sight and the line-of-contact, critical spherical latitude, center of perspective
over the North Pole).
Tangential plane at the South Pole
Tangential plane at the North Pole
sin Φ r =
R
D
=
R
R + H
versus
sin Φ r =
R
D
=
R
R + H
,
tan Φ r =
r max
R + D
=
r max
2R + H
versus
tan Φ r =
r max
H
,
r max = (2R + H) tan Φ r
versus
r max = H tan Φ r ;
(5.79)
tan x =
sin x
p
1 − sin
2 x
, tan Φ r =
R
R + H
1
q
1 −
R
(R+H) 2
=
R
p
(2R + H)H
;
(5.80)
r max = R
r
1 + 2
R
H
versus
r max =
R
q
1 + 2
R
H
.
(5.81)
5-245 The gnomonic projection
According to Fig. 5.20, the gnomonic projection is generated as a polar central perspective, where
O
∗ = O or D = 0 in the context of a normal general perspective mapping holds. In Box 5.15, the
items of such a mapping of the sphere to a plane (namely, (i) the parameterized mapping, (ii) the left
principal stretches of the left Cauchy–Green eigenspace, (iii) the left maximal angular shear, and (iv)
the inverse parameterized mapping) are collected.
Historical
aside.
Note that the gnomonic projection has been used in the antiquity for the construction of a
sundial (“gnomon”).
The basic results of the gnomonic projection or polar central perspective mapping of the sphere S
2
R
are collected in Lemma 5.4.
Lemma 5.4 (Gnomonic projection of the sphere to the polar tangential plane).
The gnomonic projection of the sphere S
2
R to the tangential plane at the North Pole is parameterized
by the two equations
x = R cot Φ cos Λ ,
y = R cot Φ sin Λ ,
(5.82)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
sin Φ
, E Φ
1
sin
2 Φ
.
(5.83)
End of Lemma.