Geography Reference
In-Depth Information
Section 15-1.
Section
15-1
offers a review of the
Korn-Lichtenstein equations
of
conformal mapping
subject
to the
integrability conditions
which are vectorial
Laplace-Beltrami equations
on a curved sur-
face, here with the metric of the ellipsoid-of-revolution. Two examples, namely UMP and UPS,
are chosen to show that the mapping equations
x
(
L, B
)and
y
(
L, B
) fulfill the Korn-Lichtenstein
equations as well as the Laplace-Beltrami equations. In addition, we present in Appendix
D
afresh
derivation of the Korn-Lichtenstein equations of conformal mapping for a (pseudo-)Riemann man-
ifold of arbitrary dimension. The standard Korn-Lichtenstein equations of conformal mapping for
a (pseudo-)Riemann manifold of arbitrary dimension extend initial results of higher-dimensional
manifolds, for instance, by
Zund
(
1987
). The standard equations of type Korn-Lichtenstein which
generate a conformal mapping of a two-dimensional Riemann manifold can be taken from standard
textbooks like
Blaschke and Leichtweiß
(
1973
)or
Heitz
(
1988
).
Section 15-2.
Section
15-2
aims at a solution of
partial differential equations
of type
Laplace-Beltrami
(second
order) as well as
Korn-Lichtenstein
(first order) in the function space of bivariate polynomials
x
(
l,b
)and
y
(
l,b
) subject to the definitions (
15.1
). The coe
cients constraints are collected in
Corollaries
15.1
and
15.2
. Note that the solution space is different from that of type separation
of variables known to geodesists from the analysis of the three-dimensional Laplace-Beltrami
equation of the gravitational potential field.
l
:=
L
−
L
0
,
(15.1)
b
:=
B
−
B
0
.
Section 15-3.
Section
15-3
, in contrast, outlines the constraints to the general solution of the Korn-Lichtenstein
equations subject to the
integrability conditions
of type
Laplace-Beltrami
, which lead directly
to the conformal coordinates of type
Gauss-Krueger
or
UTM
. Such a solution is generated by
the equidistant mapping of the
meridian of reference L
0
, for UTM up to a dilatation factor,
as the proper constraint (
x
(0
,b
)=0and
y
(0
,b
) given). The highlight is the theorem which
gives the solution of the partial differential equations for the conformal mapping in terms of a
conformal set of bivariate polynomials. Throughout, we use a right-handed coordinate system,
namely
x
“Easting” and
y
“Northing”. Boxes
15.4
and
15.5
contain the non-vanishing polynomial
coecients in a closed form.
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