Geography Reference
In-Depth Information
A
1
,A
2
.The
Laplace-Beltrami
operator “
Δ
” has a complicated structure as long as we
depend on its (
L
,
B
)-parameters. A much simpler representation of “
Δ
”isachievedassoonaswe
substitute the parameters (
L
,
B
) by the parameters (
L
,
Q
) called
isometric coordinates or confor-
mal coordinates
. Indeed the
Universal Mercator Projection
(
L, B
)
∈
E
(
L, B
)
∈
E
(
L, Q
), where
L
is already the
isometric longitude
and
Q
is the
isometric latitude
(
Lambert function
), generates
a
conformeomorphism
, simply called a
conformal map
. Section
22-3
at first presents us with a
fun-
damental solution
of the vector-valued
Laplace-Beltrami equation
A
1
,A
2
→
R
2
}
which is
not
in the class of
separation-of-variables
. Here we follow the treatment of
Grafarend
(
1995
) in representing
x
(
L, Q
)
,y
(
L, Q
) in terms of two-dimensional
harmonic Taylor polynomials
.
Secondly, we fix the unknown polynomial coecients “in function space” by a properly chosen
boundary condition which is represented in the
same function space.
Actually we postulate that
the parallel circle of reference as well as the meridian of reference is mapped by its arc length.
In addition, we unify the solution by a symmetry condition superimposed to
x
(
L, Q
)
,y
(
L, Q
)or
x
(
l,qt
)
,y
(
l,q
) subject to
l
=
L
{
Δx
(
L, Q
)=0
,Δy
(
L, Q
)=0
−
L
0
,q
=
Q
−
Q
0
. Thirdly we transform,
x
(
l,q
)
,y
(
l,q
)backto
x
(
l,q
)
,y
(
l,q
)
.l
=
L
−
L
0
,b
=
B
−
B
0
are surface normal longitude differences
L
−
L
0
and surface
normal latitude differences
B
B
0
which relate to (
L
0
,B
0
), namely reference longitude
L
0
and
reference latitude
B
0
.(
L
0
,B
0
)isthe
Gauss
choice of reference parallel
B
0
= const and reference
meridian
L
0
= const quoted from
Gauss
(
1822
). Section
22-31
is oriented towards
quality control
of the first ellipsoidal map of type “harmony”. We succeed to compute its
left principal stretches
,
namely the elements of the “
left distortion ellipse
” and present you finally via Chap.
5
with case
studies. The appendices contain the necessary analytical results, which we need in the main text.
−
22-2 Harmonic Maps
Harmonic maps are “
optimal map projections
” in the following sense. Minimize the
invariant
distortion measure
1
/
2tr
C
l
G
−
1
, namely the average (
Λ
1
+
Λ
2
)
/
2 of the principal distortion values
squared, integrated over the area to be mapped. For instance, in order to produce a series of
harmonic maps of the geodetic reference figure, the
International Reference Ellipsoid
, an ellipsoid
of revolution of semi-major axis
A
1
and semi-minor axis
A
2
, we minimize the invariant distortion
measure 1
/
2tr
C
l
G
−
1
=(
Λ
1
+
Λ
2
)
/
2 integrated over the
Reference Meridian Strip
{
L
W
≤
L
≤
L
E
,B
S
.
C
l
denotes the
left Cauchy-Green deformation tensor
(
Grafarend 1995
, pp.
431-432),
G
l
the
left metric tensor
(
Grafarend 1995
, pp. 431-432) of the ellipsoid of revolution
E
≤
B
≤
B
N
}
A
1
,A
2
, which are parameterized in terms of (GPS)
Gauss ellipsoidal longitude L
and
Gauss
ellipsoidal latitude B
. The distortion density 1
/
2tr
C
l
G
−
1
1
=(
Λ
1
+
Λ
2
)
/
2 is also called
Hilbert
invariant
of trace type. More details on
Hilbert basis invariant
,namely
Hilbert's Fundamental
Theorems
on
Invariant Theory
can be taken from
Grafarend
(
1970
). Here our geodetic analysis
of harmonic maps is based upon the following two theorems.
Theorem 22.1 (Harmonic maps, Gauss ellipsoid coordinates, variational formulation).
Let
{
x
−
x
(
L
−
B
)
,y
−
y
(
L
−
B
)
}
denote the mapping equations of Gauss ellipsoidal longitude
L
, Gauss ellipsoidal latitude.
2
(
L, B
)
∈{
(
L, B
)
∈
R
||
0
≤
L
≤
2
π,
−
π/
2
<B<
+
π/
2
}
(22.1)
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