Geography Reference
In-Depth Information
⇓
ln
f
=
n
ln tan
π
+ln
c,
Φ
2
4
−
(17.24)
α
r
=
2
n
,
nA
c
tan
4
−
(17.25)
Φ
Λ
1
=
Λ
2
=
cn
tan
4
−
2
n
Φ
.
(17.26)
R
cos
Φ
17-221 Equidistance on the Circle-of-Contact, Compare with Fig.
17.8
The constant
n
is defined using the parallel circle
Φ
=
Φ
0
which shall be mapped equidistantly,
i.e. through the cone constant
n
=sin
Φ
0
. It follows from (
17.26
)that(
17.27
)holds.
Λ
1
|
Φ
=
Φ
0
=
Λ
2
|
Φ
=
Φ
0
=
cn
tan
4
−
Φ
2
n
=1
R
cos
Φ
0
⇔
(17.27)
R
cos
Φ
0
R
cot
Φ
0
n
tan
4
−
Φ
2
n
=
tan
4
−
Φ
2
n
.
c
=
The mapping equations for this kind of projection are therefore defined through (
17.28
)or(
17.29
).
The left principal stretches are provided by (
17.30
).
⎡
n
⎤
α
r
=
Λ
sin
Φ
0
R
cot
Φ
0
tan
(
4
−
2
)
⎣
⎦
,
(17.28)
tan
(
4
−
Φ
2
)
=
R
cot
Φ
0
tan
4
−
n
2
x
y
cos(
Λ
sin
Φ
0
)
sin(
Λ
sin
Φ
0
)
,
Φ
tan
4
−
Φ
2
(17.29)
tan
4
−
n
2
Φ
Λ
1
=
Λ
2
=
cos
Φ
0
tan
4
−
Φ
2
.
(17.30)
cos
Φ
17-222 Equidistance on Two Parallels (Secant Cone,
Lambert 1772
), Compare with Fig.
17.9
The basic idea is to determine the cone constant
n
=sin
Φ
0
from an equidistant mapping of
two standard parallel circles
Φ
=
Φ
1
and
Φ
=
Φ
2
. Starting from (
17.31
), we immediately arrive
at (
17.32
), from which the integration constant
c
accordingto(
17.33
) is computed as a function
of the unknown cone constant
n
.Since
c
can also be determined via
Λ
1
|
Φ
=
Φ
2
=
Λ
2
|
Φ
=
Φ
2
:= 1, the
equality (
17.34
) is used to compute
n
accordingto(
17.35
).
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