Geography Reference
In-Depth Information
Table 22.11
Harmonic maps of the ellipsoid of revolution, band limited representation
x
=
ξ
(
q,l
)=
α
0
+
α
1
q
+
β
1
l
+
α
2
q
2
−
l
2
+2
β
2
ql
+
α
3
q
3
−
3
ql
2
++
β
3
3
q
2
l
l
3
+
−
1)
s
r
2
s
q
r−
2
s
l
2
s
++
N
r
=4
N
r
=4
α
r
[
r/
2]
s
=0
β
r
[[(
r
+1)
/
2]]
s
=1
1)
s
+1
(
−
(
−
r
2
s
q
r−
2
s
+1
l
2
s−
1
(22.78)
−
1
y
=
η
(
q,l
)=
γ
0
+
γ
1
q
+
δ
1
l
+
γ
2
q
2
−
l
2
+2
δ
2
ql
+
γ
3
q
3
−
3
ql
2
+
δ
3
3
q
2
l
l
3
−
1)
s
r
2
s
q
r−
2
s
l
2
s
+
+
N
r
γ
r
[
r/
2]
s
(
−
=4
=0
1)
s
+1
r
2
s −
q
r−
2
s
+1
l
2
s−
1
+
N
r
δ
r
[(
r
+1)
/
2]
s
(
−
(22.79)
1
=4
=1
“[
r/
2] denotes the largest natural number
≤
r/
2
,
[(
r
+1)
/
2] is the largest natural number ,
≤
(
r
+1)
/
2) respec-
tively.”
Condition #1
The
Reference Mercator L
0
is to be mapped equidistantly.
Condition #2
The parallel circle
Q
0
which passes the
Taylor point
(
Q
0
,L
0
) is gauged by the Easting coordinate
x
=
ξ
(0
,l
) to its arc length.
Condition #3
The
Easting coordinate x
=
ξ
(
q,l
) should be symmetric with respect to relative
isometric
latitude,
namely
ξ
(
−q,l
)=
ξ
(
q,l
).
Condition #4
In contrast, the Northing coordinate
y
=
η
(
q,l
) should be symmetric with respect to
isometric
longitude,
namely
η
(
q,l
)=
η
(
q,
l
).
While the
Reference Meridian L
0
is mapped
equidistantly,
the parallel circle
Q
0
is
not
.This
result can be seen by inspecting
−
for the special case
q
= 0 in Eqs.(
22.78
)
and (
22.79
)ofTable
22.11
.
x
=
ξ
(0
,l
)
,y
=
η
(0
,l
) contains nonlinear term of
L
{
ξ
(
q,l
)
,η
(
q,l
)
}
L
0
:=
l
.
Obviously the parallel circle
Q
0
,
q
=0,is
not
mapped onto a straight line in contrast to the
Reference Meridian L
0
,l
= 0 which is mapped equidistantly as a
straight line
in the chart
{
−
}
Let us define the
equidistant mapping of the Reference Meridian L
0
and the special mapping
of the
Reference Parallel Circle Q
0
in more detail.
x
=
ξ
(
q,l
)
,y
=
η
(
q,l
)
22-33 The First Boundary Condition or the Equidistant Mapping
of the Reference Meridian
The postulate of an equidistant mapping of the Reference meridian accounts for the computation
of the arc length of the elliptic meridian according to Eq.(
22.80
)ofTable
22.12
. Indeed following
the theory of Taylor polynomials we have to decompose the arc length into the
zero order part
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