Geography Reference
In-Depth Information
22
Optimal Map Projections by Variational Calculus:
Harmonic Maps
Harmonic maps
are a certain kind of an
optimal map projection
which has been developed for
map projections of the
sphere
. Here we generalize it to the “
ellipsoid of revolution
”. The subject
of an
optimization
of a map projection is not new for a cartographer. For instance, in Sect.
5-25
,
we compute the
minimumdistortion energy
for mapping the “
sphere-to-plane
”. For maps of type
(i) orthographic, (ii) conformal (UPS), (iii) gnomonic, (iv) equiareal (Lambert), (v) equidistant
(Postel) and (vi) Lagrange conformal we computed the “
best maps
”byorderingoftype(
5.134
)
and (
5.135
). Another example is the
optimal cylinder projection of the sphere
in comparing (i)
conformal maps, (ii) equiareal maps and (iii) distance preserving maps with respect to optimal of
type
Airy
and
Airy-Kavrajski
. Section
10-3
,Figs.
10.7
and
10.8
lists the optimal. A final example is
the optimal design of the
Universal Transverse Mercator
Projection (UTM), namely the
optimal
dilatation factor
. Section
15-4
, Examples
15.3
and
15.4
as well as Fig.
15.5
lists the result.
Harmonic maps are generated as a certain class of optimal map projections. For instance,
if the distortion energy over a Meridian Strip of the
International Reference Ellipsoid
is min-
imized we are led to the
Laplace-Beltrami vector-valued partial differential equation
.Herewe
construct harmonic functions
x
(
L, B
)
,y
(
L, B
) given as functions of ellipsoidal surface param-
eters (
L, B
)oftype
{
Gauss ellipsoidal longitude
L
, Gauss ellipsoidal latitude
B
}
as well as
x
(
l,q
)
,y
(
l,q
) given as functions of relative isometric longitude
l
=
L
−
L
0
and relative isomet-
ric latitude
q
=
Q
−
Q
0
gauged to a vector-valued boundary condition of special symmetry.
{
of the new harmonic map is given in Tables
22.18
and
22.21
. The distortion energy analysis of the new harmonic map is presented as well as case stud-
ies for (i)
B
Easting, Northing
}
or
{
x
(
b, l
)
,y
(
b, l
)
}
40
◦
,
+40
◦
]
,L
31
◦
,
+49
◦
]
,B
0
=
30
◦
,L
0
=9
◦
and
(ii)
B
[46
◦
,
56
◦
]
,L
∈
[
−
∈
[
−
±
∈
∈
[4
.
5
◦
,
7
.
5
◦
];[7
.
5
◦
,
10
.
5
◦
];[10
.
5
◦
,
13
.
5
◦
];[13
.
5
◦
,
16
.
5
◦
]
,B
0
=51
◦
,L
0
6
◦
,
9
◦
,
12
◦
,
15
◦
}
{
}
∈{
22-1 Introduction
Harmonic maps
are generated as a certain class of optimal map projections: Minimize the distor-
tion energy over a closed and bounded surface or of a part like the
Meridian Strip
(in general, a
two-dimensional
Riemann manifold
) to find the
Laplace-Beltrami
vector-valued partial differential
equation as the
Euler-Lagrange equations
by means of variational calculus. Those harmonic func-
tions
x
(
L, B
)
,y
(
L, B
) given as a function of surface parameters (
L, B
) specified later generate
a harmonic map in the sense of
{Δx
=0
,Δy
=0
}
where“
Δ
”
}
is the two-dimension
Laplace-
Beltrami
differential operator. Here we aim at generating a harmonic map
x
(
L, B
)
,y
(
L, B
)ofthe
ellipsoid of revolution more specifically the
International Reference Ellipsoid
as the representative
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