Authors:
L. Battha Geodetic and Geophysical Research Institute of the Hungarian Academy of Sciences POB 5 H-9401 Sopron Hungary

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J. Závoti Geodetic and Geophysical Research Institute of the Hungarian Academy of Sciences POB 5 H-9401 Sopron Hungary

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In a basic problem of geodesy the directions from points with known coordinates to an unknown (new) point are measured, and then the resulting angles are used to compute the coordinates of the new point. The relations between angles and lengths lead to a system of nonlinear equations of the form fi = 0 ( i = 1, 2, 3), where each fi is a second degree polynomial of the unknown distances x1 , x2 , x3 . Two different direct (non-iterative) solutions are discussed: one is based on the Sylvesterdeterminant of the resultant (this is a new result), the other on the Gröbner-bases. We show that in the general case both methods lead to the same equations in one variable and of fourth degree, but in a special case the equations obtained from Sylvester-determinant are of second degree. As a numerical example, three known points and an unknown point were selected in the city of Sopron. The required space angles were used to make the computations yielding the X, Y, Z coordinates of the unknown point.We show that the direct solution of the 2D similarity transformation leads to the same result as applying the Gröbner-bases.

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Acta Geodaetica et Geophysica
Language English
Size B5
Year of
Foundation
2013
Volumes
per Year
1
Issues
per Year
4
Founder Magyar Tudományos Akadémia
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó Springer
Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 2213-5812 (Print)
ISSN 2213-5820 (Online)

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