172 5 “Sphere to tangential plane”: polar (normal) aspect
Box 5.6 (Equiareal mapping of the sphere to the tangential plane at the North Pole).
Postulate of a areomorphism:
Λ 1 Λ 2 = 1 ,
f (∆)
R 2 sin ∆
df (∆)
d∆
= 1 ⇒ f (∆)df (∆) = R
2 sin ∆d∆ .
(5.37)
Integration of the characteristic differential equation of an equiareal mapping S
2
R → T N S
2
R
subject to an initial condition:
2
4
f
2
2
= −R
2 cos ∆ + c
r(0) = f (0) = 0
3
5 ⇒ 0 = −R
2 + c ⇒ c = R
2 ,
2
6
4
f
2 = 2R
2 (1 − cos ∆) = 4R
2 sin
2 ∆
2
cos x = 1 − 2 sin
2 x
2
3
7
5 ⇒ r = f (∆) = 2R sin
∆
2
.
(5.38)
Parameterized equiareal mapping:
"
x
y
#
= 2R sin
∆
2
"
cos Λ
sin Λ
#
= 2R sin
„
π
4
−
Φ
2
« "
cos Λ
sin Λ
#
.
(5.39)
Left principal stretches:
f
(∆) = R cos
∆
2
,
Λ 1 =
1
cos
∆
2
=
1
cos
` π
4
−
Φ
2
´ , Λ 2 = cos
∆
2
= cos
„
π
4
−
Φ
2
«
;
(5.40)
special value (isometry): Φ →
π
2
: lim
Φ→π/2
Λ 1 = lim
Φ→π/2
Λ 2 = 1 .
(5.41)
Left eigenvectors:
C 1 Λ 1 = E Λ
1
cos
` π
4
−
Φ
2
´
(“Easting”) ,
C 2 Λ 2 = E Φ cos
„
π
4
−
Φ
2
«
(“Northing”) .
(5.42)
Left maximal angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
1 − cos
2 ∆
2
1 + cos 2 ∆
2
.
(5.43)
Parameterized inverse mapping:
tan Λ =
y
x
,
sin
∆
2
=
1
2R
p
x 2 + y 2 .
(5.44)