314 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
This two-step procedure has at least two basic disadvantages. On the one hand, it is in general
difficult to set up a first set of conformal coordinates. For instance, due to the involved difficulties the
Philosophical Faculty of the University of Goettingen Georgia Augusta dated 13th June 1857 set up
the “Preisaufgabe” to find a conformal mapping to the triaxial ellipsoid. Based upon Jacobi’s contribution on elliptic coordinates (C. G. J. Jacobi 1839) the ”Preisschrift” of E. Schering (1857) was finally
crowned, nevertheless leaving the numerical problem open as how to construct a conformal map of the
triaxial ellipsoid of type UTM. For an excellent survey, we refer to W. Klingenberg (1982), H. Schmehl
(1927), recently, to B. Mueller (1991). There is another disadvantage of the two-step procedure. The
equivalence between two-dimensional real-valued Riemann manifolds and one-dimensional complexvalued manifolds holds only for analytic Riemann manifolds. In E. Grafarend (1995), we give two
counterexamples of surfaces of revolution which are from the differentiability class C
∞ , but which are
not analytical. Accordingly, the theory of holomorphic functions does not apply. Finally, one encounters great difficulties in generalizing the theory of conformal mappings to higher-dimensional (pseudo-)
Riemann manifolds. Only for even-dimensional (pseudo-)Riemann manifolds of analytic type, multidimensional complex analysis can be established. We experience a total failure for odd-dimensional
(pseudo-)Riemann manifolds as they appear in the theory of refraction, Newton mechanics, or plumb
line computation, to list just a few conformally flat three-dimensional Riemann manifolds. The theory of conformal mapping took quite a different direction when A. Korn (1914) and L. Lichtenstein
(1911, 1916) set up their general differential equations for two-dimensional Riemann manifolds, which
govern conformality. They allow the straightforward transformation of ellipsoidal coordinates of type
surface normal longitude L and latitude B into conformal coordinates of type Gauss-Krueger or UTM
(x, y) without any intermediate conformal coordinate system of type UMP or UPS! Accordingly, our
objective here is a proof of our statement!
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 surface, 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 a fresh derivation of the Korn–
Lichtenstein equations of conformal mapping for a (pseudo-)Riemann manifold 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 J. 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 W. Blaschke and
K. Leichtweiss (1973) or S. 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 coefficients constraints are collected in Corollary 15.1 and
Corollary 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 ,
b := B − B 0 .
(15.1)
This two-step procedure has at least two basic disadvantages. On the one hand, it is in general
difficult to set up a first set of conformal coordinates. For instance, due to the involved difficulties the
Philosophical Faculty of the University of Goettingen Georgia Augusta dated 13th June 1857 set up
the “Preisaufgabe” to find a conformal mapping to the triaxial ellipsoid. Based upon Jacobi’s contribution on elliptic coordinates (C. G. J. Jacobi 1839) the ”Preisschrift” of E. Schering (1857) was finally
crowned, nevertheless leaving the numerical problem open as how to construct a conformal map of the
triaxial ellipsoid of type UTM. For an excellent survey, we refer to W. Klingenberg (1982), H. Schmehl
(1927), recently, to B. Mueller (1991). There is another disadvantage of the two-step procedure. The
equivalence between two-dimensional real-valued Riemann manifolds and one-dimensional complexvalued manifolds holds only for analytic Riemann manifolds. In E. Grafarend (1995), we give two
counterexamples of surfaces of revolution which are from the differentiability class C
∞ , but which are
not analytical. Accordingly, the theory of holomorphic functions does not apply. Finally, one encounters great difficulties in generalizing the theory of conformal mappings to higher-dimensional (pseudo-)
Riemann manifolds. Only for even-dimensional (pseudo-)Riemann manifolds of analytic type, multidimensional complex analysis can be established. We experience a total failure for odd-dimensional
(pseudo-)Riemann manifolds as they appear in the theory of refraction, Newton mechanics, or plumb
line computation, to list just a few conformally flat three-dimensional Riemann manifolds. The theory of conformal mapping took quite a different direction when A. Korn (1914) and L. Lichtenstein
(1911, 1916) set up their general differential equations for two-dimensional Riemann manifolds, which
govern conformality. They allow the straightforward transformation of ellipsoidal coordinates of type
surface normal longitude L and latitude B into conformal coordinates of type Gauss-Krueger or UTM
(x, y) without any intermediate conformal coordinate system of type UMP or UPS! Accordingly, our
objective here is a proof of our statement!
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 surface, 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 a fresh derivation of the Korn–
Lichtenstein equations of conformal mapping for a (pseudo-)Riemann manifold 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 J. 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 W. Blaschke and
K. Leichtweiss (1973) or S. 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 coefficients constraints are collected in Corollary 15.1 and
Corollary 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 ,
b := B − B 0 .
(15.1)