Geography Reference
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the
left surface element
dS
l
=
dL dQ
det
G
l
=
dL dQA
2
(
Q
)=
dl dqA
2
(
Q
)
(22.52)
has been represented in terms of
isometric coordinates of Mercator type
(
L, Q
).
End of Theorem.
For the proof of Theorem
22.2
we refer to Appendix 2. Let us comment here on the results. By
means of Eq.(
22.41
) we define the domain of the
Reference Meridian Strip. L
0
denotes the
Ref-
erence Longitude
like in Gauss-Krueger conformal mapping or in
Universal Transverse Mercator
Projection
(UTM). Since the isometric coordinates (
L, Q
) are generating a conformal mapping of
Mercator type,
the distortion energy takes the special from with respect to the
left factor of confor-
mality Λ
2
(
Q
) as outlined in Eqs.(
22.42
)-(
22.43
). In particular the surface element of Eq.(
22.52
)
is specific to the
left factor of conformality Λ
2
(
Q
) of type Eq.(
22.50
). The transformation of
Gauss surface normal latitude B to isometric latitude
Q
is parameterized by Eq.(
22.51
). The
coordinates of the
left metric tensor
G
l
are given in the conformal from of Eq.(
22.48
), while the
coordinates of the
left Cauchy-Green deformation tensor
C
l
are specialized by Eq.(
22.49
). Finally
enjoy the simple structure of the
Laplace-Beltrami equation
in isometric coordinates (conformal
coordinates, isothermal coordinates).
Now let us solve the
Laplace-Beltrami
equation in isometric coordinates of Mercator type on
the ellipsoid-of-revolution.
22-3 Solving the Laplace-Beltrami Equation on the International
Reference Ellipsoid, the Boundary Value Problem of an Arc
Preserving Mapping
Here we have two targets. At first we are going to solve the
Laplace-Beltrami equation
which
generates harmonic functions on the ellipsoid of revolution which represents the
International
Reference Ellipsoid.
Secondly based upon such a solution in the function space of
Taylor polyno-
mials
we determine the coecients of such a harmonic expansion by means of a properly chosen
vector-valued
boundary value problem
.
22-31 Solving the Vector-Valued Laplace-Beltrami Equation
in the Function Space of Harmonic Taylor Polynomials
How should we solve the
Laplace-Beltrami
equation
Δξ
(
q,l
)=0
,Δη
(
q,l
) = 0 which generates
harmonic maps x
=
ξ
(
q,l
)
,y
=
η
(
q,l
)=0
One standard method of solving
Δξ
=0
,Δη
= 0 is by means of
separation of variables
as
outlined in
Grafarend
(
1995
, pp. 452-454) for harmonic functions on the ellipsoid of revolution.
Alternatively, following a proposal of
Gauss
(
1822
)wesetup
harmonic Taylor polynomials
which
do
not
follow the recipe of separation of variables.
In Table
22.4
, we present you with the “ansatz” for bivariate harmonic Taylor polynomials
in relative isometric coordinates of
Mercator type.
We setup by means of Eqs.(
22.54
)-(
22.57
)
homogeneous, bivariate polynomials ξ
(
q,l
)
,η
(
q,l
) of degree r (rank r) which are supposed to
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