Geography Reference
In-Depth Information
figure of the Earth. In this case (
L
,
B
) as a pair of ellipsoidal coordinates refer to “surface nor-
mal” ellipsoidal longitude
L
and “surface normal” ellipsoidal latitude
B
. Such a specific set of
parameters derives its name from its spherical image of the ellipsoidal surface normal, called
“
Gauss map
”(
Grafarend 2000
). Indeed it is the standard
geodetic coordinate system
used in
LPS (“Local Positioning System”: local problem solver) and GPS (“Global Positioning Sys-
tem”: global problem solver). In order to solve the vector-valued
Laplace-Beltrami
equation
Δx
(
L, B
)=0
,Δy
(
L, B
)=0
,
(
L, B
)
A
1
,A
2
→
R
2
(
x, y
)
uniquely
we have to formulate a
vector-valued
boundary value problem
. Its solution leads us to the
new map
of the ellipsoid of
revolution, which is in “
harmony
”.
There is
vast literature
on harmonic maps. Any advanced textbook on differential geometry,
differential topology or variational calculus contains a review on harmonic maps. Let us mention:
Wood
(
1994
),
Melko and Stirling
(
1994
),
Yang
(
2000
) focus on the concepts of
distortion energy
and tension
and give an invariant formulation.
Hamilton
(
1975
) gives the
general Laplacian gen-
erating
harmonic maps. As examples of harmonic maps the following list is given: (i) harmonic
maps are the harmonic functions, (ii) harmonic maps are geodesics, (iii) every isometry is har-
monic, (iv) a conformal map is one, which preserves angles, every conformal map is harmonic,
(the converse is
not
true) and others. give an existence theorem of harmonic mappings of weak
type.
Jorgens
(
1955
) explored harmonic mappings between double connected regions including iso-
lated singularities. In a special volume
Jost
(
1964
) addresses the theoretical problem of harmonic
maps between surfaces by focusing on conformal mappings, the minimizing distortion energy in a
restricted subclass of a special Sobolev Space on the related Dirichlet problem, in particular the
existence theorem of harmonic maps between compact surfaces by
Lemaire
. Furthermore he dis-
cusses the
Ruh-Vilms Theorem
stating that the
Gauss map
of a submanifold of a
Euclidean space
with constant mean curvature is harmonic as well as immersed surfaces of
constant Gauss curva-
ture
,in
∈
E
3
.Ofmore
constructive nature
is
Jost
(
1995
) text where harmonic maps are developed
on 108 pages.
Harmonic maps beside their use in map projections found great applications in
General Rel-
ativity.
On the level of the first post-Newtonian approximation (1pN) the
harmonic gauge
has
been introduced for generating “
harmonic coordinates
”. For more details let us refer to
Damour
et al.
(
1991
), (pp. 75-76),
Schwarze
(
1995
, pp. 18-21) and
Weinberg
(
1972
, pp. 161-163, 179, 181,
216-220, 254, 262).
Actually we wonder that for optimal map projections harmonic maps have not yet been used
for geodetic maps of the Earth. Probably the
missing boundary value problem
may be considered
as an argument for such a situation as already been pointed out by
Grafarend
(
1995
)inthereview
paper on optimal
Universal Transverse Mercator Projection
(opt UTM). Another reason may be
that
ellipsoidal harmonic map
subject to a properly chosen boundary value problem is
neither
conformal
nor
equilateral, but of minimal distortion energy. Such a harmony gives us hope that
there is a bright future of the
ellipsoidal harmonic map
for “mapping the Earth”.
Here we are organized as following: In Sect.
22-2
we set up the functional of
type energy
(average
distortion energy) by means of a properly chosen
energy density
(distortion density) (tr
C
l
G
−
1
/
2),
which is integrated over the
reference meridian strip
.
C
l
denotes the
left Cauchy-Green defor-
mation tensor
and
G
l
, respectively, the left metric tensor of
2
A
1
,A
2
, the ellipsoid of revolution
of semi-major axis
A
1
and semi-minor axis
A
2
. Indeed such an energy density is the arithmetic
mean of the
lift principle stretches
(
Λ
1
,
Λ
2
), namely
Λ
1
+
Λ
2
)
/
2, which build up the “
left distortion
ellipse
”
E
A
1
,A
2
.
In consequence, we outline the
first variation
of energy functional which leads us to the vector-
valued
Laplace-Beltrami
equations
{Δx
(
L, B
)
,Δy
(
L, B
)
}
in terms of surface normal coordinates
E
Search WWH ::
Custom Search