5-2 Special mapping equations 179
Box 5.10 (General normal perspective mapping of the sphere to the equatorial plane of reference).
Parameterized mapping (polar coordinates):
α = Λ , r =
D
R sin Φ + D
R cos Φ =
R + H
R(1 + sin Φ) + H
cos Φ .
(5.60)
Special case H = 0:
r = R tan
„
π
4
−
Φ
2
«
= R tan
∆
2
.
(5.61)
Parameterized mapping (Cartesian coordinates):
"
x
y
#
=
2
6
6
4
D
R sin Φ + D
R cos Φ cos Λ
D
R sin Φ + D
R cos Φ sin Λ
3
7
7
5 =
2
6
6
6
4
R + H
R(1 + sin Φ) + H
R cos Φ cos Λ
R + H
R(1 + sin Φ) + H
R cos Φ sin Λ
3
7
7
7
5
,
"
x
y
#
=
D
R sin Φ + D
R cos Φ
"
cos Λ
sin Λ
#
=
R + H
R(1 + sin Φ) + H
R cos Φ
"
cos Λ
sin Λ
#
.
(5.62)
Special case H = 0:
"
x
y
#
= R tan
„
π
4
−
Φ
2
« "
cos Λ
sin Λ
#
= R tan
∆
2
"
cos Λ
sin Λ
#
.
(5.63)
Left principal stretches:
Λ 1 =
f (∆)
R sin ∆
, Λ 2 =
f
(∆)
R
;
(5.64)
Λ 1 =
D
R sin Φ + D
=
R + H
R(1 + sin Φ) + H
,
Λ 2 = D
R + D sin Φ
(R sin Φ + D) 2 = (R + H)
R(1 + sin Φ) + H sin Φ
[R(1 + sin Φ) + H]
2
.
(5.65)
Special case H = 0:
Λ 1 = Λ 2 =
1
1 + sin Φ
=
1
1 + cos ∆
, Λ 1 = Λ 2 =
1
2
1
cos 2 ∆
2
=
1
2
1
cos 2
` π
4
−
Φ
2
´ .
(5.66)
Box 5.10is a summary of the general normal perspective mapping of the sphere S
2
R to the equatorial
plane of reference, specifically of the parameterized mapping in both polar coordinates {α, r} and in
Cartesian coordinates {x, y}, completed by the computation of the left principal stretches {Λ 1 , Λ 2 }.
For the special case O
∗ = S or, equivalently, D = R or H = 0 , we prove conformality Λ 1 = Λ 2 . For such
a configuration of the southern perspective center, Φ = −π/2 is singular: Λ 1 (−π/2)= Λ 2 (π/2) → ∞.