Geography Reference
In-Depth Information
Postulate.
The polar coordinate
r
depends only on the ellipsoidal latitude
Φ
or on the ellipsoidal colatitude
Δ
:=
π/
2
Φ
, i.e.
r
=
x
2
+
y
2
=
f
(
Δ
)=
f
(
π/
2
−
−
Φ
). If
Φ
=
π/
2 or, equivalently,
Δ
=0,then
f
(0) = 0 holds.
End of Postulate.
In last consequence, the general equations of an azimuthal mapping are provided by the following
vector equation:
x
y
=
r
cos
α
=
f
(
Δ
)cos
Λ
.
(8.12)
r
sin
α
f
(
Δ
)sin
Λ
Question: “How can we identify the images of the special
coordinate lines
Λ
= constant and
Φ
= constant, respec-
tively?” Answer (
y
=
x
tan
Λ, Λ
= constant : elliptic
meridian): “The image of the elliptic meridian
Λ
=con-
stant under an azimuthal mapping is the radial straight
line.” Answer (
x
2
+
y
2
=
r
2
=
f
2
(
Δ
)
,Δ
= constant :
parallel circle): “The image of the parallel circle
Δ
=con-
stant (or
Φ
= constant) under an azimuthal mapping is
the circle
1
r
of radius
r
=
f
(
Δ
). Such a mapping is called
concircular
.”
S
Proof (
y
=
x
tan
Λ, Λ
= constant : elliptic meridian).
Solve the first equation towards
f
(
Δ
)=
x/
cos
Λ
and substitute
f
(
Δ
) in the second equation such
that
y
=
f
(
Δ
)sin
Λ
=
x
sin
Λ/
cos
Λ
=
x
tan
Λ
holds.
End of Proof (
y
=
x
tan
Λ, Λ
= constant : elliptic meridian).
Proof (
x
2
+
y
2
=
r
2
=
f
2
(
Δ
)
,Δ
= constant : parallel circle).
Compute the terms
x
2
and
y
2
and add the two:
x
2
+
y
2
=
f
2
(
Δ
)
.
End of Proof (
x
2
+
y
2
=
r
2
=
f
2
(
Δ
)
,Δ
= constant : parallel circle).
In summary, the images of the elliptic meridian and the parallel circle constitute the typical
graticule of an azimuthal mapping, i.e.
meridians(
Λ
= constant)
−→
radial straight lines
,
parallel circles
Δ
= constant
(8.13)
−→
equicentric circles
.
Φ
= constant
Box
8.2
shows a collection of formulae which describe the left Jacobi matrix J
l
as well as the
left Cauchy-Green matrix C
l
for an azimuthal mapping
2
2
O
E
A
1
,A
2
→
P
. The left pair of matrices
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