Solution of the intersection problem by the Sylvester-resultant and a comparison of two solutions of the 2D similarity transformation

Battha, L. and Závoti, J. (2009) Solution of the intersection problem by the Sylvester-resultant and a comparison of two solutions of the 2D similarity transformation. Acta Geodaetica et Geophysica Hungarica, 44 (4). pp. 429-438. ISSN 1217-8977

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Abstract

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 Gr¨obner-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 Gr¨obner-bases.

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