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Orthogonal Distance Fit of a Pyramid (Geodäsie/Vermessung)

MichaeL ⌂, Bad Vilbel, Monday, 10.12.2012, 18:31 (vor 4233 Tagen) @ Balata

Hi,

Can I ask, if you could summarize or describe the algorithm "pyramid" in a logical steps, in order to understand it. Because, some steps still I couldn't understand it.

You are referred to the listed publications for details to the least-squares algorithm.

To fit the pyramid, the algorithm estimates the parameters of the 4 planes. The planes are parametrized as Hesse normal form:

n_xx_i + n_yy_i + n_zz_i = d

with the secondary condition
||n|| \stackrel{!}{=} \sqrt{n_x^2 + n_y^2 + n_z^2} \stackrel{!}{=} 1

For each plane the unknowns are n_x, n_y, n_z and d. In your case, the algorithm estimates 4×4 = 16 plane-parameters. Up to this point, the algorithm produces the same results as a single fit of each plane.

To estimate the intersection, the number of unknowns increased by 3 - the coordinates of the intersection-point. To force a intersection of the 4 planes, 4 additional conditions are introduced. If K is the intersection-point, each plane has to contain K, or similar:

n_xx_K + n_yy_K + n_zz_K \stackrel{!}{=} d

This is the whole mathematic model, which can be used in a common orthogonal distance fit algorithm - see listed publications to get formulas of Gauß-Helmert model.

kind regads
Micha

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applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences


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