A new method of transforming Cartesian to geodetic (or planetographic) coordinates on a triaxial ellipsoid is presented. The method is based on simple reasoning coming from essentials of vector calculus. The reasoning results in solving a nonlinear system of equations for coordinates of the point being the projection of a point located outside or inside a triaxial ellipsoid along the normal to the ellipsoid. The presented method has been compared to a vector method of Feltens (J Geod 83:129-137, 2009) who claims that no other methods are available in the literature. Generally, our method turns out to be more accurate, faster and applicable to celestial bodies characterized by different geometric parameters. The presented method also fits to the classical problem of converting Cartesian to geodetic coordinates on the ellipsoid of revolution.
A new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 -10 km) (terrestrial), (20 -1000 km), and (1000 -36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σ ϕ < 0.0004") and height (σ h < 10 −6 km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.
The paper presents results of the transformation between two height systems Kronstadt’60 and Kronstadt’86 within the area of Krakow’s district, the latter system being nowadays a part of National Spatial Reference System in Poland. The transformation between the two height systems was carried out based on the well known and frequently applied in geodesy polynomial regression. Despite the fact it is well known and frequently applied it is rather seldom broader tested against the optimal degree of a polynomial function, goodness of fit and its predictive capabilities. In this study some statistical tests, measures and techniques helpful in analyzing a polynomial transformation function (and not only) have been used.
We present a method of approximation of a deformation eld based on the local a ne transformations constructed based on n nearest neighbors with respect to points of adopted grid. The local a ne transformations are weighted by means of inverse distance squared between each grid point and observed points (nearest neighbors). This work uses a deformation gradient, although it is possible to use a displacement gradient instead -the two approaches are equivalent. To decompose the deformation gradient into components related to rigid motions (rotations, translations are excluded from the deformation gradient through di erentiation process) and deformations, we used a polar decomposition and decomposition into a sum of symmetric and an anti-symmetric matrices (tensors). We discuss the results from both decompositions. Calibration of a local a ne transformations model (i.e., number of nearest neighbors) is performed on observed points and is carried out in a cross-validation procedure. Veri cation of the method was conducted on simulated data-grids subjected to known (functionally generated) deformations, hence, known in every point of a study area.
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