5-2 Special mapping equations 181
By means of Box 5.12, we have collected the parameter equations which characterize the general
normal perspective mapping of the sphere S
2
R to the southern tangential plane, specifically in terms
of polar coordinates {α, r} and of Cartesian coordinates {x, y}, completed by the computation of the
left principal stretches {Λ 1 , Λ 2 }. At the point of symmetry, namely ∆ = π or Φ = −π/2, we prove the
isometry Λ 1 = Λ 2 = 1.
Box 5.12 (General normal perspective mapping of the sphere to the tangential plane at minimal distance).
Parameterized mapping (polar coordinates):
α = Λ ,
r = (D − R)
R cos Φ
D − R |sin Φ|
=
H
H + R(1 − |sin Φ|)
R cos Φ .
(5.71)
Parameterized mapping (Cartesian coordinates):
"
x
y
#
=
2
6
6
4
(D − R)
R cos Φ cos Λ
D − R |sin Φ|
(D − R)
R cos Φ sin Λ
D − R |sin Φ|
3
7
7
5 ,
"
x
y
#
=
H
H + R(1 − |sin Φ|)
R cos Φ
"
cos Λ
sin Λ
#
.
(5.72)
Left principal stretches:
Λ 1 =
f (∆)
R sin ∆
,
Λ 2 =
f
(∆)
R
;
(5.73)
Λ 1 =
D − R
D − R |cos ∆|
=
H
H + R(1 − |sin Φ|)
,
Λ 2 = (D − R)
D |cos ∆| − R
(D − R |cos ∆|) 2 = H
H − R(1 − |sin Φ|)
[H + R(1 − |sin Φ|)] 2 .
(5.74)
Special isometry:
∆ → π (|cos ∆| → 1) or Φ → −π/2 (|sin Φ| → 1)
⇒
Λ 1 = Λ 2 = 1 .
(5.75)
5-244 Line-of-sight, line-of-contact, minimal and complete atlas
The line-of-sight as well as the line-of-contact for both the general normal perspective mapping to the
tangential plane at the North Pole and to the tangential plane at the South Pole are illustrated in
Fig. 5.13 and Fig. 5.14, respectively.