Geography Reference
In-Depth Information
15
“Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect
Mapping the ellipsoid-of-revolution to a cylinder: transverse aspect. Transverse Mercator pro-
jection, Gauss-Krueger/UTM coordinates. Korn-Lichtenstein equations, Laplace-Beltrami
equations.
Conventionally, conformal coordinates, also called conformal charts , representing the surface of
the Earth or any other Planet as an ellipsoid-of-revolution , also called the Geodetic Reference
Figure , are generated by a two-step procedure. First, conformal coordinates (isometric coordi-
nates, isothermal coordinates) of type UMP ( Universal Mercator Projection , compare with Exam-
ple 15.1 )oroftype UPS ( Universal Polar Stereographic Projection , compare with Example 15.2 )
are derived from geodetic coordinates such as surface normal ellipsoidal longitude/ellipsoidal lat-
itude. UMP is classified as a conformal mapping on a circular cylinder, while UPS refers to a
conformal mapping onto a polar tangential plane with respect to an ellipsoid-of-revolution, an
azimuthal mapping. The conformal coordinates of type UMP or UPS, respectively, are conse-
quently complexified, just describing the two-dimensional Riemann manifold of type of ellipsoid-
of-revolution as one-dimensional complex manifold .Namely,the real-valued conformal coordinates
x and y of type UMP or UPS, respectively, are transformed into the complex-valued conformal
coordinate z = x +i y . Second, the conformal coordinates ( x, y )
z of type UMP or UPS,
respectively, are transformed into another set of conformal coordinates, called Gauss-Krueger
or UTM , by means of holomorphic functions w ( z )( w := u +i v
) with respect to complex
algebra and complex analysis. Indeed, holomorphic functions directly fulfill the d ' Alembert-Euler
equations ( Cauchy-Riemann equations ) of conformal mapping as outlined by Grafarend ( 1995 ),
for instance. Consult Figs. 15.1 and 15.2 for a first impression.
This two-step procedure has at least two basic disadvantages. On the one hand, it is in gen-
eral dicult to set up a first set of conformal coordinates. For instance, due to the involved
diculties 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 ellip-
soid. Based upon Jacobi's contribution on elliptic coordinates ( Jacobi 1839 ) the “Preisschrift” of
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
Klingenberg ( 1982 ), Schmehl ( 1927 ), recently, to Mueller ( 1991 ). There is another disadvantage
of the two-step procedure. The equivalence between two-dimensional real-valued Riemann mani-
folds and one-dimensional complex-valued manifolds holds only for analytic Riemann manifolds.
C
Search WWH ::




Custom Search