Geography Reference
In-Depth Information
S
r
, meta-pole coordinates
L
0
= 180
◦
and
Fig. 16.1.
Universal Oblique Mercator Projection of the sphere
30
◦
. Compare with Fig.
16.2
B
0
=
−
Section 16-1.
In particular, in Sect.
16-1
, we review the fundamental equations which govern conformal map-
ping of a two-dimensional Riemann manifold, namely (i) the
Korn-Lichtenstein equations
, (ii)
the
Laplace-Beltrami equations
(the integrability conditions of the Korn-Lichtenstein equations),
and (iii) the condition
preserving the orientation
of a conformeomorphism, for the ellipsoid-
of-revolution
parameterized by ellipsoidal longitude
L
and ellipsoidal latitude
B
.Two
examples for the solution of the fundamental equations (i), (ii), and (iii) are given, namely (1)
the
Universal Mercator Projection
(UMP), and (2) the
Universal Polar Stereographic Projection
(UPS). If the equations (i), (ii), and (iii) of a conformeomorphism are specialized to UMP or
UPS as input conformal coordinates, the equations for output conformal coordinates of another
type are obtained as (
α
) the d'Alembert-Euler equations (the Cauchy-Riemann equations), (
β
)
the Laplace-Beltrami equations (the integrability conditions of the d'Alembert-Euler equations),
(
γ
) the condition
preserving the orientation
of a conformeomorphism. A fundamental solution of
the equations (
α
), (
β
), and (
γ
) is given in the class of homogeneous polynomials and interpreted
with respect to the two-dimensional conformal group C
6
(2) constituted by six parameters (two
for translation, one for rotation, one for dilatation, two for special conformal) embedded in the
two-dimensional conformal group C
∞
(2), which is described by infinite set of parameters.
E
A
1
,A
2
Section 16-2.
2
A
1
,A
2
Section
16-2
introduces the oblique reference frame of
E
, in particular, the oblique meta-
2
equator
E
a
,b
which is parameterized by reduced meta-longitude
α
. Section
16-3
determines the
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