Summary
Riemann 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 [-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 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.
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References
Airy, G. B. (1861): Explanation of a projection by balance of errors for maps applying to a very large extent of the earth's surface and comparison of this projection with other projections, Phil. Mag.22 (1861) 409–421
Beltrami, E. (1866): Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette, Annali di Matematica7 (1866) 185–204
Boltz, H. (1943): Formeln und Tafeln zur numerischen Berechnung Gauß-Krügerscher Koordinaten aus den geographischen Koordinaten, Veröff. Geod. Inst. Potsdam, Neue Folge 111, Potsdam 1943
Fialkow, A. (1939): Conformal geodesics, Trans. Amer. Math. Soc.45 (1939) 443–473
Gauss, C. F. (1827): Disquisitiones generales circa superficies curvas (1827), Werke 4, 217–258, deutsch von Wangerin, Ostwalds Klassiker, Bd. 5
Grafarend, E. W. (1994): The Optimal Universal Transverse Mercator Projection, manuscripta geodaetica (1994)
Grafarend, E. W. and P. Lohse (1991): The minimal distance mapping of the topographic surface onto the (reference) ellipsoid of revolution, manuscripta geodaetica (1991)
Grafarend, E. W. and R. J. You (1994): The Newton form of a geodesic in Maupertuis gauge on the sphere and the biaxial ellipsoid, Zeitschrift für Vermessungswesen (1994)
Legendre, A. M. (1806): Analyse des triangles tracés sur la surface d'un sphéroide. Tome VII de la 1°-série des memoires de l'Académie des Sciences, Paris 1806
Lichtenegger, H. (1987): Zur numerischen Lösung geodätischer Hauptaufgaben auf dem Ellipsoid, Zeitschrift für Vermessungswesen112 (1987) 506–515
Riemann, G. F. B. (1851): Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe, Inauguraldissertation, Göttingen 1851
Schouten, J. A. (1954): Ricci calculus, Springer Verlag, Berlin 1954
Vogel, W. O. (1970): Kreistreue Transformationen in Riemannschen Räumen, Arch. Math.21 (1970) 641–645
Vogel, W. O. (1973): Einige Kennzeichnungen der homothetischen Abbildungen eines Riemannschen Raumes unter den kreistreuen Abbildungen, manuscripta mathematica9 (1973) 211–228
Weingarten, J. (1861): Über eine Klasse auf einander abwickelbarer Flächen, J. f. reine u. angew. Math.59 (1861) 382–393
Yano, K. (1979): On Riemannian manifolds admitting an infinitesimal conformal transformation, Math. Z.113 (1970) 205–214
Yano, K. (1940, 1942): Concircular geometry I-V, Proc. Imperial Academy16 (1940) 195–200, 354–360, 442–448, 505–511,18 (1942) 446–451
Yano, K. (1940): Conformally separable quadratic differential forms, Proc. Imp. Acad. Tokyo16 (1940) 83-
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Grafarend, E.W., Syffus, R. The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates. Journal of Geodesy 70, 13–37 (1995). https://doi.org/10.1007/BF00863416
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DOI: https://doi.org/10.1007/BF00863416