Geography Reference
In-Depth Information
= 0, the image of the North Pole is a circular arc. The final mapping equations now result
to (
17.10
)or(
17.11
) with the left principal stretches (
17.12
). The circle-of-contact,
Φ
=
Φ
0
,is
mapped equidistantly and conformally, i.e.
Λ
1
|
Φ
=
Φ
0
=
Λ
2
|
Φ
=
Φ
0
=1.
α
r
Since
c
=
Λ
sin
Φ
0
,
(17.10)
R
(
Φ
0
−
Φ
+cot
Φ
0
)
x
y
=
R
(
Φ
0
−
Φ
+cot
Φ
0
)
cos(
Λ
sin
Φ
0
)
,
(17.11)
sin(
Λ
sin
Φ
0
)
Λ
1
=
sin
Φ
0
(
Φ
0
−
Φ
+cot
Φ
0
)
cos
Φ
,
(17.12)
Λ
2
=1
.
Fig. 17.5.
Mapping the sphere to a cone. Polar aspect, equidistant mapping of the set of meridians, equidistant
and conformal on the standard parallel
Φ
=
Φ
0
=30
◦
(Ptolemy projection)
17-212 Equidistance and Conformality on the Circle-of-Contact, Point-Like Image of the North
Pole, Compare with Fig.
17.6
As a special case of the Ptolemy projection the equidistant mapping with point-like pole is
obtained by setting the integration constant
c
to zero. The mapping Eqs.(
17.14
)or(
17.15
)and
the left principal stretches (
17.16
) are easily derived from Eq.(
17.7
).
Λ
1
|
Φ
=
Φ
0
,c
=0
=
nR
(
π/
2
Φ
0
)
R
cos
Φ
0
−
=1
⇒
(17.13)
cos
Φ
0
n
=
π/
2
− Φ
0
,
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