226 8 “Ellipsoid-of-revolution to tangential plane”
Box 8.3 (Equidistant mapping of the ellipsoid-of-revolution to the tangential plane at the North Pole).
Parameterized mapping:
α = Λ , r = f (∆), ∆ := π/2 − Φ ,
x = r cos α = f (∆) cos Λ , y = r sin α = f (∆) sin Λ .
(8.20)
Canonical postulate Λ 2 = 1,
equidistant mapping of the family of meridians:
Λ 2 = f
(∆)
(1 − E
2 cos
2 ∆)
3/2
A 1 (1 − E 2 )
= 1
⇔
df = A 1 (1 − E
2 )
d∆
(1 − E 2 cos 2 ∆) 3/2
⇒
f (∆) = A 1 (1 − E
2 )
Z ∆
0
d∆
(1 − E 2 cos 2 ∆) 3/2 .
(8.21)
Transformation of surface normal latitude Φ to reduced latitude Φ
∗ :
tan Φ
∗ =
p
1 − E 2 tan Φ
⇔
tan Φ =
1
√
1 − E 2
tan Φ
∗ .
(8.22)
Equidistant mapping of the family of meridians,
elliptic integral of the second kind:
f (∆) → f (Φ) ,
f (Φ) = A 1 (1 − E
2 )
Z π/2
π/2−Φ
dΦ
(1 − E 2 sin
2 Φ ) 3/2 ;
f (Φ) → f (Φ
∗ ) ,
f (Φ
∗ ) = A 1
Z π/2
π/2−Φ ∗
p
1 − E 2 cos 2 Φ ∗ dΦ
∗ ;
f (Φ
∗ ) → f (∆
∗ ) ,
f (∆
∗ ) = A 1
Z ∆
∗
0
p
1 − E 2 sin
2 ∆d∆ =: A 1 E(∆
∗ , E) .
(8.23)
Elliptic integral of the second kind:
f (Φ) = A 1 E
h
π/2 − arc tan
“ p
1 − E 2 tan Φ
”
, E
i
.
(8.24)