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The laws of nature in general, and the relations and laws in geodesy in particular can be expressed in most cases by nonlinear equations which are in general solved by transforming them to linear form and applying iteration. The process of bringing the equations to linear form implies neglections and approximation. In certain cases it is possible to obtain exact, correct solutions for nonlinear problems. In the present work we introduce parameters into the rotation matrix, and using this we derive solutions for the 2D and 3D similarity transformations. This method involves no iteration, and it does not require the transformation of the equations to linear form. The scale parameter is determined in both cases by solving a polynomial equation of second degree. This solution is already known, but our derivation is worth consideration because of its simple nature.

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The closed form solution of 7 parameter 3D transformation.The Gauss-Jacobi combinatorial adjustment is applied to solve the 3D transformation problem with 7 parameters, and it is also demonstrated that the combinatorial algorithm gives the same solution as the conventional linear Gauss-Markov model.

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