Geography Reference
In-Depth Information
C
l
=J
l
G
r
J
l
=
n
2
f
2
f
2
.
0
(17.4)
0
Λ
1
=
C
11
R
cos
Φ
, Λ
2
=
C
22
G
22
=
f
nf
G
11
=
R
.
(17.5)
17-2 Special Mapping Equations
Setting up special equations of the mapping “sphere to cone”. Equidistant, conformal, and
equal area mappings. Ptolemy, de L'Isle, Lambert, and Albers projections. Point-like North
Pole.
17-21 Equidistant Mapping (de L'Isle Projection)
The general mapping equations for this type of mappings are derived from the postulate (
17.6
)
such that (
17.7
) holds. The integration constant
c
has to be determined from the additional
requirement that the image of the North Pole is a point or a circular arc. Setting
c
=0,a
point-like image of the North Pole is attained.
Λ
2
=
f
2
−
Φ
f
π
Φ
=
R
f
π
Φ
=
R
π
Φ
+
c,
=1
⇔
2
−
⇒
2
−
2
−
(17.6)
R
α
r
=
,
nΛ
R
2
−
Φ
+
c
Λ
1
=
n
R
2
−
Φ
+
c
R
cos
Φ
,Λ
2
=1
.
(17.7)
17-211 Equidistance and Conformality on the Circle-of-Contact (C. Ptolemy, 85-150
ad
),
Compare with Fig.
17.5
We require the circle-of-contact to be mapped equidistantly and thus state (
17.8
)fromwhich—
together with the cone constant
n
=sin
Φ
0
—the integration constant
c
is determined as (
17.9
).
Λ
1
|
Φ
=
Φ
0
=
n
R
2
−
Φ
0
+
c
=1
,
(17.8)
R
cos
Φ
0
c
=
R
cos
Φ
0
2
+
Φ
0
=
R
cot
Φ
0
−
2
+
Φ
0
.
π
π
n
−
(17.9)
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