Rainer Syffus

An algorithm for the inverse of a multivariate homogeneous polynomial of degreen

SummaryVarious geodetic problems (the free nonlinear geodetic boundary value problem, the computation of Gauß-Krüger coordinates or UTM coordinates, the problem of nonlinear regression) demand theinversion of an univariate, bivariate, trivariate, in generalmultivariate homogeneous polynomial of degree n. The new algorithm which is oriented towardsSymbolic Computer Manipulation is based upon the algebraic power base computation with respect toKronecker-Zehfuβ product structure leading to the solution of a system oftriangular matrix equations: Only the first row of the inverse triangular matrix has to be computed. TheSymbolic Computer Manipulation program of the GKS algorithm is available from the authors.

The oblique azimuthal projection of geodesic type for the biaxial ellipsoid : Riemann polar and normal coordinates

SummaryRiemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the “reference” ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (“Lagrange portrait”) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (“Legendre series”) and in theHamilton portrait (“Lie series”).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (Λ1, Λ2 = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gauβ-Krüger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-lE,lE] = [-2°, +2°], [bS,bN] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gauβ-Krüger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others.

From Riemann Manifolds to Riemann Manifolds

Mappings from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold, simultaneous diagonalization of two matrices, mappings (isoparametric, conformal, equiareal, isometric, equidistant), measures of deformation (Cauchy–Green deformation tensor, Euler–Lagrange deformation tensor, stretch, angular shear, areal distortion), decompositions (polar, singular value), equivalence theorems of conformal and equiareal mappings (conformeomorphism, areomorphism), Korn–Lichtenstein equations, optimal map projections.

“Sphere to Tangential Plane”: Polar (Normal) Aspect

Mapping the sphere to a tangential plane: polar (normal) aspect. Equidistant, conformal, and equal area mappings. Normal perspective mappings. Pseudo-azimuthal mapping. Wiechel polar pseudo-azimuthal mapping. Northern tangential plane, equatorial plane, southern tangential plane. Gnomonic and orthographic projections. Lagrange projection.

“Sphere to Tangential Plane”: Transverse Aspect

Mapping the sphere to a tangential plane: meta-azimuthal projections in the transverse aspect. Equidistant, conformal (stereographic), and equal area (transverse Lambert) mappings.

“Sphere to Cone”: Pseudo-Conic Projections

Mapping the sphere to a cone: pseudo-conic projections. The Stab–Werner mapping and the Bonne mapping. Tissot indicatrix.

“Sphere to Cone”: Polar Aspect

Mapping the sphere to a cone: polar aspect. Equidistant, conformal, and equal area mappings. Ptolemy, de L’Isle, Lambert, and Albers projections. Point-like North Pole. Tangent cones, secant cones, and circles-of-contact.

“Ellipsoid-of-Revolution to Cone”: Polar Aspect

Mapping the ellipsoid-of-revolution \(\mathbb{E}_{A_{1},A_{2}}^{2}\) to a cone: polar aspect. Lambert conformal conic mapping and Albers equal area conic mapping.

From Riemann Manifolds to Euclidean Manifolds

Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold, Cauchy–Green and Euler–Lagrange deformation tensors, equivalence theorem for equiareal mappings, conformeomorphism and areomorphism, Korn–Lichtenstein equations and Cauchy–Riemann equations, Mollweide projection, canonical criteria for (conformal, equiareal, isometric, equidistant) mappings, polar decomposition and simultaneous diagonalization for more than two matrices.

Surfaces of Gaussian Curvature Zero

Classification of surfaces of Gaussian curvature zero (Gauss flat, two-dimensional Riemann manifolds) in a two-dimensional Euclidean space, ruled surfaces, developable surfaces.