5-2 Special mapping equations 177
Box 5.8 (General normal perspective mapping of the sphere to the tangential plane at maximal distance).
Parameterized mapping (polar coordinates):
α = Λ ,
r =
R + D
R sin Φ + D
R cos Φ =
1 +
R
D
1 +
R
D
sin Φ
R cos Φ .
(5.49)
Parameterized mapping (Cartesian coordinates):
"
x
y
#
=
2
6
6
4
R + D
R sin Φ + D
R cos Φ cos Λ
R + D
R sin Φ + D
R cos Φ sin Λ
3
7
7
5 =
2
6
6
6
6
4
1 +
R
D
1 +
R
D
sin Φ
R cos Φ cos Λ
1 +
R
D
1 +
R
D
sin Φ
R cos Φ sin Λ
3
7
7
7
7
5
,
"
x
y
#
=
R + D
R sin Φ + D
R cos Φ
"
cos Λ
sin Λ
#
=
1 +
R
D
1 +
R
D
sin Φ
R cos Φ
"
cos Λ
sin Λ
#
.
(5.50)
Left principal stretches:
Λ 1 =
f (∆)
R sin ∆
, Λ 2 =
f
(∆)
R
;
(5.51)
Λ 1 =
R + D
R cos ∆ + D
=
1 +
R
D
1 +
R
D
cos ∆
,
Λ 2 =
R + D
(R cos ∆ + D) 2 (D cos ∆ + R) =
„
1 +
R
D
«
R
D
+ cos ∆
`
1 +
R
D
cos ∆
´ 2 ,
(5.52)
or
Λ 1 =
R + D
R sin Φ + D
=
1 +
R
D
1 +
R
D
sin Φ
,
Λ 2 =
R + D
(R sin Φ + D) 2 (D sin Φ + R) =
„
1 +
R
D
«
R
D
+ sin Φ
`
1 +
R
D
sin Φ
´ 2 .
(5.53)
Special isometry:
∆ → 0 or Φ → π/2
⇒
Λ 1 = Λ 2 = 1 .
(5.54)
Box 5.8is a summary of the general normal perspective mapping of the sphere S
2
R to the northern
tangential plane, 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 }.
At the point of symmetry, namely ∆ = 0or Φ = π/2, we prove the isometry Λ 1 = Λ 2 .