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Message:

> Hello Mark0,
> 
> > I've got a few questions about the confidence ellipses I get in JAG3D. When adjusting 2D network I get a (major semi axis) and c (minor semi axis) with gamma as a rotation angle of the confidence ellipse.
> 
> Yes, that is right.
> 
> > But once I take the same project (and add a height data) and do 3D network adjustment I get the same results when it comes to the coordinates, but I also get b (middle semi axis) and two more rotation angles that I didn't have in 2D network. Now when I look at the numbers I see that b and c are very close in values that I had as a and c before. What is the meaning of major semi axis in this case then?
> > And none of the rotation angles are close or similar to one I had before in 2D network adjustment. So how would I draw one for example in CAD software when having 3 rotation angles? What do they represent?
> 
> Each dispersion matrix (covariance matrix) is [link=https://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizable_matrices]diagonalizable[/link] using, for instance, the [link=https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix]eigendecomposition[/link]. Appliying the eigendecomposition results in a diagonal matrix, which contains the so-called eigenvalues, and an [link=https://en.wikipedia.org/wiki/Orthogonal_matrix]orthogonal matrix[/link], which contains the corresponding eigenvectors. In case of a spatial network, a 3x3 sub-matrix of the total dispersion matrix  refers to each estimated point. Decomposing a 3x3 dispersion yields three eigenvalues and three eigenvectors forming a 3x3 orthogonal matrix. The eigenvalues relate to the semi-axis of the confidence region, which is represented by an [link=https://en.wikipedia.org/wiki/Ellipsoid]ellipsoid[/link] in case of a spatial network. (In a horizontal network, the confidence region forms an ellipse. In case of a levelling network, the confidence region is just an interval.) The eigenvectors are the direction vectors of these semi-axes. The eigenvectors forms a 3x3 orthogonal matrix. If the determinate of this orthogonal matrix is +1, it is also a [link=https://en.wikipedia.org/wiki/Rotation_matrix]rotation matrix[/link]. In case of a horizontal network, a single Euler angle represents the rotation. However, in case of a spatial network, three Euler-angles are required to describe a full rotation sequence. For more details, please take a look to the [link=https://software.applied-geodesy.org/wiki/least-squares-adjustment/reliability#genauigkeitsmasse_der_parameter]documentation[/link].
> 
> All the best
> Micha

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