Identification of an ellipse and a sub-ellipse of errors by elements of the covariance matrix

Abstract

The authors present a study of the point position mistake relationship with the elements of the error ellipse. The comparison of the mentioned error, determined by the F. R. Helmert’s and P. Werkmeister’s formulas, is given. It is shown that they both are related to the elements of the error ellipse and the covariance matrix of the coordinate vector. For the conditions of the polar serif, the dependence of the correlation coefficient of the determined point coordinates on the directional angle of the line to this point, as well as that of the angular and linear measurements mutual accuracy, is established. The relationships between the elements of the covariance matrix and those of the ellipse of point position errors are investigated for various special cases. Namely, when the correlation coefficient is zero, it is ± 1, in the case of equality of the determined point’s inverse weight matrix diagonal elements and when the determination of the point’s position was performed by a polar serif. The error ellipse dimension is considered as a function of linear COEX systems in all directions. Its graphs and subtypes are presented depending on the determined point’s inverse weight matrix elements values