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π
2
− Φ
cos
Φ
,Λ
2
=1
.
Δ
sin
Δ
=
Λ
1
=
(5.19)
Left eigenvectors:
Δ
sin
Δ
(Easting)
, C
2
Λ
2
=
E
Φ
(Northing)
.
C
1
Λ
1
=
E
Λ
(5.20)
Parameterized inverse mapping:
tan
Λ
=
y
x
, Δ
=
x
2
+
y
2
.
(5.21)
R
2
Left maximal angular distortion:
Ω
l
=2arcsin
=2arcsin
Λ
1
−
Λ
2
Λ
1
+
Λ
2
Δ
sin
Δ
Δ
+sin
Δ
−
.
(5.22)
Lemma 5.1 (EPM, equidistant mapping of the sphere to the tangential plane at the North Pole).
The equidistant mapping of the sphere to the tangential plane at the North Pole, in short, EPM
(Equidistant Polar Mapping), is parameterized by
x
=
RΔ
cos
Λ
=
R
π
2
− Φ
cos
Λ, y
=
RΔ
sin
Λ
=
R
π
2
− Φ
sin
Λ,
(5.23)
Δ
sin
Δ
,
E
Φ
subject to the left Cauchy-Green eigenspace
{
E
Λ
}
.
End of Lemma.
Fig. 5.4.
A spherical vertical section, an equidistant mapping of the sphere to the tangential plane at the North
Pole
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