Geography Reference
In-Depth Information
α
r
=
Λ
cos
Φ
0
,
π/
2
−Φ
0
R
(
π/
2
(17.14)
−
Φ
)
⎡
cos
Λ
cos
Φ
0
π/
2
−Φ
0
⎤
x
y
=
R
(
π
⎣
sin
Λ
cos
Φ
0
⎦
,
2
−
Φ
)
(17.15)
π/
2
−
Φ
0
Λ
1
=
cos
Φ
π/
2
−
Φ
0
π/
2
− Φ
, Λ
2
=1
.
(17.16)
cos
Φ
0
Fig. 17.6.
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
◦
, point-like North Pole
17-213 Equidistance and Conformality on Two Parallels (Secant Cone, J.N. de L'Isle
1688-1768), Compare with Fig.
17.7
If instead of one parallel two parallel circles are required to be mapped equidistantly, this approach
leads to a secant cone, the so-called
de L'Isle projection
, named after the French astronomer
Joseph Nicolas de L'Isle. We start from (
17.7
) and demand that (
17.17
) is satisfied for the two
parallel circles
Φ
=
Φ
1
and
Φ
=
Φ
2
. We obviously receive two equations for the two unknowns
n
:= sin
Φ
0
(cone constant!) and
c
, the result of which is (
17.18
). We end up with the mapping
equations (
17.19
)or(
17.20
) with the left principal stretches (
17.21
). For
Φ
=
Φ
1
or
Φ
=
Φ
2
,we
even experience conformality (isometry),
Λ
1
=
Λ
2
=1.
Λ
1
|
Φ
=
Φ
1
=
n
R
2
−
Φ
1
+
c
R
cos
Φ
1
=
Λ
1
|
Φ
=
Φ
2
=
n
R
2
−
Φ
2
+
c
R
cos
Φ
2
=1
,
(17.17)
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