5-2 Special mapping equations 195
Box 5.17 (Lagrange projection).
Parameterized special perspective mapping (polar coordinates):
α = Λ ,
r = R tan
∆
2
= R tan
„
π
4
−
Φ
2
«
.
(5.100)
Sine lemma (triangle OpS):
r
sin
∆
2
=
R
cos
∆
2
→ r .
(5.101)
Parameterized special perspective mapping (Cartesian coordinates):
"
x
y
#
= R tan
∆
2
"
cos Λ
sin Λ
#
= R tan
„
π
4
−
Φ
2
« "
cos Λ
sin Λ
#
.
(5.102)
Case (i) (Φ = 0):
"
x
y
#
= R
"
cos Λ
sin Λ
#
.
(5.103)
Case (ii) (Φ = π/2):
"
x
y
#
=
"
0
0
#
.
(5.104)
Left principal stretches:
Λ 1 =
f
(∆)
R sin ∆
=
tan
∆
2
2 sin
∆
2
cos
∆
2
=
1
2 cos 2 ∆
2
,
Λ 2 =
f
(∆)
R
=
1
2 cos 2 ∆
2
.
(5.105)
Conformality:
Λ 1 = Λ 2 .
(5.106)
Left eigenvectors:
C 1 Λ 1 = E Λ
1
2 cos 2 ∆
2
(Easting) ,
C 2 Λ 2 = E Φ
1
2 cos 2 ∆
2
(Northing) .
(5.107)