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The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates
Authors:E W Grafarend  R Syffus
Institution:(1) Department of Geodetic Science, Stuttgart University, Keplerstr. 11, D-70174 Stuttgart, Germany
Abstract:Summary Riemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the ldquoreferencerdquo 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 (ldquoLagrange portraitrdquo) 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 (ldquoLegendre seriesrdquo) and in theHamilton portrait (ldquoLie seriesrdquo).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 (Lambda1, Lambda2 = 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 Gaubeta-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 -l E,l E] = -2°, +2°], b S,b N] = -2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gaubeta-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.
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