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In-Depth Information
l
r
, commented references
Tab l e 1 . 4
Conformal mapping
M
→
M
Two-dimensional conformal mapping, Korn-Lichtenstein equations:
n
=2.
Blaschke and Leichtweiß
(
1973
)
Bourguignon
(
1970
)
n
-dimensional conformal mapping, Weyl curvature:
n ≥
3.
Eisenhart
(
1949
)
n
-dimensional conformal mapping, Weyl curvature:
n
= arbitrary.
Finzi
(
1922
)
Three-dimensional conformal mapping, generalized Korn-Lichtenstein equations:
n
=3.
Gauss
(
1822
)
Classical contribution on two-dimensional conformal mapping:
n
=2.
Gauss
(
1816-1827
-
1827
)
Classical contribution on conformal mapping of the ellipsoid-of-revolution.
Grafarend and Syffus
(
1998c
)
n
-dimensional conformal mapping, generalized Korn-Lichtenstein equations.
Hedrick and Ingold
(
1925a
)
Analytic functions in three dimensions.
Hedrick and Ingold
(
1925b
)
Laplace-Beltrami equations in three dimensions.
Klingenberg
(
1982
)
Conformal mapping of the triaxial ellipsoid, elliptic coordinates.
Kulkarni
(
1969
)
Curvature structures and conformal mapping.
Kulkarni
(
1972
)
Conformally flat manifolds.
Kulkarni and Pinkall
(
1988
)
Conformal geometry.
Lafontaine
(
1988a
,
b
)
Conformal mapping “from the Riemann viewpoint”.
Liouville
(
1850
)
Three-dimensional conformal mapping.
Ricci
(
1918
)
Conformal mapping.
Schouten
(
1921
)
n
-dimensional conformal mapping, Weyl curvature.
Weyl
(
1918
)
Conformal mapping.
Weyl
(
1921
)
Conformal mapping.
Yanushaushas
(
1982
)
Three-dimensional conformal mapping, generalized Korn-Lichtenstein equations.
Zund
(
1987
)
Three-dimensional conformal mapping, generalized Korn-Lichtenstein equations.
Example 1.11 (Conformal mapping of an ellipsoid-of-revolution
E
A
1
,A
1
,A
2
toasphere
S
r
:the
Universal Mercator Projection (UMP) of type left
E
r
, the special Korn-
Lichtenstein equations, and the Cauchy-Riemann equations (d'Alembert-Euler equations)).
A
1
,A
1
,A
2
and right
S
Let us assume that we have found a solution of the left Korn-Lichtenstein equations of the
ellipsoid-of-revolution
2
A
1
A
1
,A
2
E
parameterized by the two coordinates
{
Λ, Φ
}
which conventionally
are called
{
Gauss surface normal longitude, Gauss surface normal latitude
}
. Similarly, let us
2
depart from a solution of the right Korn-Lichtenstein equations of the sphere
S
r
parameterized
by the two coordinates
. Here, we
follow the commutative diagram of Fig.
1.25
and identify the left conformal coordinates
{
λ, φ
}
which are called
{
spherical longitude, spherical latitude
}
{
P, Q
}
2
Λ
1
,A
1
,A
2
with the Universal Mercator Projection (UMP) of
E
, and the right conformal coordinates
2
{
p, q
}
with the Universal Mercator Projection (UMP) of
S
r
, which is outlined in Box
1.24
.The
ratios
Q/A
1
and
q/r
are also called
or
ellipsoidal spherical
Lambert functions Q
=
A
1
lam
Φ
and
q
=
r
lam
φ
, respectively. In addition,
we adopt the left and right matrices of the metric
{
ellipsoidal isometric latitude, spherical isometric latitude
}
{
G
l
,
G
r
}
of Example
1.3
.
End of Example.
We pose four problems. (i) Do the left and right conformal maps that are parameterized by
{
as “UMP left” and “UMP right” fulfil the Korn-Lichtenstein equa-
tions, the integrability conditions (vector-valued Laplace-Beltrami equations of harmonicity), and
the condition “orientation preserving conformeomorphism”? (ii) Derive the left and right factors
of conformality,
Λ
2
=
Λ
1
=
Λ
2
and
λ
2
=
λ
1
=
λ
2
. Do the factors of conformality fulfill a special
Helmholtz equation? (iii) Prove that under “UMP left” as well as “UMP right” both the equators
of
P
(
Λ
)
,Q
(
Φ
)
}
and
{
p
(
λ
)
,q
(
φ
)
}
2
A
1
,A
1
,A
2
2
r
are mapped equidistantly. Interpret this result as a boundary condition of the
Korn-Lichtenstein equations. (iv) Derive a “simple conformal mapping”
E
and
S
2
2
E
A
1
A
1
,A
2
→
S
r
.
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