VIII Preface
Geodesics and geodetic mappings, in particular, the geodesic circle, the Darboux frame, and the
Riemann polar and normal coordinates, are the topic of Chapter 20. We illustrate the Lagrange and the
Hamilton portrait of a geodesic, introduce the Legendre series, the corresponding Hamilton equations,
the notion of initial and boundary value problems, the Riemann polar and normal coordinates,
Lie series, and specialize to the Clairaut constant and to the ellipsoid-of-revolution. Geodetic parallel coordinates refer to Soldner coordinates. Finally, we refer to Fermi coordinates. The deformation
analysis of Riemann, Soldner, and Gauss–Krueger coordinates is presented.
Datum problems.
Datum problems, namely its analysis versus synthesis and its Cartesian approach versus curvilinear
approach, are presented in Chapter 21. Examples reach from the transformation of conformal coordinates of type Gauss–Krueger and type UTM from a local datum (regional, national, European) to a
global datum (WGS 84) of type UM (Universal Mercator).
Appendices.
Appendix A is entitled as “Law and order”. It brings up relation preserving maps. We refer to Venn
diagrams, Euler circles, power sets, Hesse diagrams, finally to fibering. The inversion of univariate,
bivariate, in general, multivariate homogeneous polynomials is presented in Appendix B. In contrast,
Appendix C reviews elliptic functions and elliptic integrals. Conformal mappings are the key subject of Appendix D. First, we treat the classical Korn–Lichtenstein equations. Second, we treat the
celebrated d’Alembert–Euler equations (usually called Cauchy–Riemann equations) which generate
both conformal mapping, (i) on the the basis of real algebra and (ii) on the basis of complex algebra. Lemma D.1 gives three alternative formulations of the Korn–Lichtenstein equations. The fundamental solutions of the d’Alembert–Euler equations subject to the harmonicity condition is reviewed in Lemma D.2 in terms of a polynomial representation (D.15)–(D.29). An alternative solution
in terms of matrix notation based upon the Kronecker–Zehfuss product is provided by (D.30) and
(D.31). Lemmas D.3 and D.4 review two solutions of the d’Alembert–Euler equations subject to the
integrability conditions of harmonicity, by separation of variables this time. Two choices of solving
the basic equations of the transverse Mercator projection are presented: x = x(q, p), y = y(q, p). We
especially estimate (i) the boundary condition for the universal transverse Mercator projection modulo an unknown dilatation factor and (ii) we solve the already formulated boundary value problem
with respect to the d’Alembert–Euler equations (Cauchy–Riemann equations). Finally, the unknown
dilatation factor is optimally determined by optimizing the total distance distortion measure (Airy
optimum) or the total areal distortion. Appendix E introduces the extrinsic terms geodetic curvature,
geodetic torsion, and normal curvature, the notion of a geodesic circle, especially the Newton form
of a geodesic in Maupertuis gauge on the sphere and on the ellipsoid-of-revolution. Mixed cylindrical maps of the ellipsoid-of-revolution of type equiareal based upon the Lambert projection and the
sinusoidal Sanson–Flamsteed projection, especially as the horizontal weighted mean versus the vertical weighted mean, are the central topics of Appendix F. The generalized Mollweide projection and
the generalized Hammer projection (generalized for the ellipsoid-of-revolution) are the key topics, especially of our studies in Appendix G and Appendix H. The optimal Mercator projection and the
optimal polycylindric projection of type conformal, here developed on the ellipsoid-of-revolution, are
applied to the many islands of the Indonesian Archipellagos in Appendix I. Projection heights in the
geometry space are the topic of Appendix J. We treat the plane, the sphere, the ellipsoid-of-revolution,
and the triaxial ellipsoid, and we review the solution algorithm for inverting Cartesian coordinates
to projection heights. An example is the Buchberger algorithm. In detail, we review surface normal
coordinates, for example, in the computation of the triaxial ellipsoids of type Earth, Moon, Mars,
Phobos, Amalthea, Io, and Mimas.
Geodesics and geodetic mappings, in particular, the geodesic circle, the Darboux frame, and the
Riemann polar and normal coordinates, are the topic of Chapter 20. We illustrate the Lagrange and the
Hamilton portrait of a geodesic, introduce the Legendre series, the corresponding Hamilton equations,
the notion of initial and boundary value problems, the Riemann polar and normal coordinates,
Lie series, and specialize to the Clairaut constant and to the ellipsoid-of-revolution. Geodetic parallel coordinates refer to Soldner coordinates. Finally, we refer to Fermi coordinates. The deformation
analysis of Riemann, Soldner, and Gauss–Krueger coordinates is presented.
Datum problems.
Datum problems, namely its analysis versus synthesis and its Cartesian approach versus curvilinear
approach, are presented in Chapter 21. Examples reach from the transformation of conformal coordinates of type Gauss–Krueger and type UTM from a local datum (regional, national, European) to a
global datum (WGS 84) of type UM (Universal Mercator).
Appendices.
Appendix A is entitled as “Law and order”. It brings up relation preserving maps. We refer to Venn
diagrams, Euler circles, power sets, Hesse diagrams, finally to fibering. The inversion of univariate,
bivariate, in general, multivariate homogeneous polynomials is presented in Appendix B. In contrast,
Appendix C reviews elliptic functions and elliptic integrals. Conformal mappings are the key subject of Appendix D. First, we treat the classical Korn–Lichtenstein equations. Second, we treat the
celebrated d’Alembert–Euler equations (usually called Cauchy–Riemann equations) which generate
both conformal mapping, (i) on the the basis of real algebra and (ii) on the basis of complex algebra. Lemma D.1 gives three alternative formulations of the Korn–Lichtenstein equations. The fundamental solutions of the d’Alembert–Euler equations subject to the harmonicity condition is reviewed in Lemma D.2 in terms of a polynomial representation (D.15)–(D.29). An alternative solution
in terms of matrix notation based upon the Kronecker–Zehfuss product is provided by (D.30) and
(D.31). Lemmas D.3 and D.4 review two solutions of the d’Alembert–Euler equations subject to the
integrability conditions of harmonicity, by separation of variables this time. Two choices of solving
the basic equations of the transverse Mercator projection are presented: x = x(q, p), y = y(q, p). We
especially estimate (i) the boundary condition for the universal transverse Mercator projection modulo an unknown dilatation factor and (ii) we solve the already formulated boundary value problem
with respect to the d’Alembert–Euler equations (Cauchy–Riemann equations). Finally, the unknown
dilatation factor is optimally determined by optimizing the total distance distortion measure (Airy
optimum) or the total areal distortion. Appendix E introduces the extrinsic terms geodetic curvature,
geodetic torsion, and normal curvature, the notion of a geodesic circle, especially the Newton form
of a geodesic in Maupertuis gauge on the sphere and on the ellipsoid-of-revolution. Mixed cylindrical maps of the ellipsoid-of-revolution of type equiareal based upon the Lambert projection and the
sinusoidal Sanson–Flamsteed projection, especially as the horizontal weighted mean versus the vertical weighted mean, are the central topics of Appendix F. The generalized Mollweide projection and
the generalized Hammer projection (generalized for the ellipsoid-of-revolution) are the key topics, especially of our studies in Appendix G and Appendix H. The optimal Mercator projection and the
optimal polycylindric projection of type conformal, here developed on the ellipsoid-of-revolution, are
applied to the many islands of the Indonesian Archipellagos in Appendix I. Projection heights in the
geometry space are the topic of Appendix J. We treat the plane, the sphere, the ellipsoid-of-revolution,
and the triaxial ellipsoid, and we review the solution algorithm for inverting Cartesian coordinates
to projection heights. An example is the Buchberger algorithm. In detail, we review surface normal
coordinates, for example, in the computation of the triaxial ellipsoids of type Earth, Moon, Mars,
Phobos, Amalthea, Io, and Mimas.