234 8 “Ellipsoid-of-revolution to tangential plane”
The Equidistant Polar Mapping (EPM) of the ellipsoid-of-revolution is summarized in Lemma 8.10,
which is based upon the direct mapping equations, its left principal stretches, the left eigenvectors,
the left maximal angular distortion, and the inverse mapping equations that are collected in Box 8.5.
Lemma 8.10 (Equidistant Polar Mapping (EPM), equidistant mapping of the ellipsoid-of-revolution to
the tangential plane at the North Pole).
The equidistant mapping of the spheroid to the tangential plane at the North Pole of an oblate
ellipsoid-of-revolution, in short, Equidistant Polar Mapping (EPM), is parameterized by
x = f (∆) cos Λ , y = f (∆) sin Λ ,
(8.44)
subject to the left Cauchy–Green eigenspace {E Λ Λ 1 (∆), E Φ }. The radial function r = f (∆) that
represents the meridian arc length from the North Pole to a point on the meridian Λ = constant is
given either in the form of an elliptic integral of the second kind or in the series expansion of Box 8.5.
End of Lemma.
Box 8.5 (Equidistant mapping of the ellipsoid-of-revolution to the tangential plane at the North Pole).
Parameterized mapping:
α = Λ , r = f (∆) , ∆ := π/2 − Φ , f(∆) → f (Φ) ,
x = r cos α = f (Φ) cos Λ , y = r sin α = f (Φ) sin Λ .
(8.45)
Series expansion, equidistant mapping of the family of meridians:
f (Φ) = A 1
»
E 0
“ π
2
− Φ
”
− E 2 sin 2Φ − E 4 sin 4Φ − E 6 sin 6Φ − E 8 sin 8Φ − E 10 sin 10Φ + O(E
12 )
–
. (8.46)
Parameterized equidistant mapping:
x = A 1 E 0
“ π
2
− Φ
”
cos Λ−
−A 1
`
E 2 sin 2Φ + E 4 sin 4Φ + E 6 sin 6Φ + E 8 sin 8Φ + E 10 sin 10Φ + O(E
12 )
´
cos Λ ,
y = A 1 E 0
“ π
2
− Φ
”
sin Λ−
−A 1
`
E 2 sin 2Φ + E 4 sin 4Φ + E 6 sin 6Φ + E 8 sin 8Φ + E 10 sin 10Φ + O(E
12 )
´
sin Λ .
(8.47)
Left principal stretches and left eigenvectors:
Λ 1 =
f (∆)
√
1 − E 2 cos 2 ∆
A 1 sin ∆
=
f (Φ)
p
1 − E 2 sin
2 Φ
A 1 cos Φ
, Λ 2 = 1 ,
(8.48)
C 1 = E Λ =
D Λ X
D Λ X
(“Easting”) , C 2 = E Φ =
D Φ X
D Φ X
= −E ∆ (“Northing”) ,
(i) C 1 Λ 1 = E Λ
f (Φ)
p
1 − E 2 sin
2 Φ
A 1 cos Φ
, (ii) C 2 Λ 2 = E Φ = −E ∆ .
(8.49)
Left angular distortion:
d l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
˛
f (Φ)
p
1 − E 2 sin
2 Φ − A 1 cos Φ
f (Φ)
p
1 − E 2 sin
2 Φ + A 1 cos Φ
˛
˛
˛
˛
˛
.
(8.50)
Parameterized inverse mapping, Λ = α, tan Λ = y/x (r =
p
x 2 + y 2 ):
Φ =
π
2
−
r
A 1 E 0
− F 2 sin 2
r
A 1 E 0
− F 4 sin 4
r
A 1 E 0
− F 6 sin 6
r
A 1 E 0
− F 8 sin 8
r
A 1 E 0
−
−F 10 sin 10
r
A 1 E 0
+ O(E
12 ) .
(8.51)