Explicit solution of the overdetermined three-dimensional resection problem

Abstract.

 Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the overdetermined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of the Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are then reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ³ is generated via the Error Propagation law/variance–covariance propagation. The Fast Nonlinear Adjustment Algorithm was proposed by C.F. Gauss, whose work was published posthumously, and C.G.I. Jacobi. The algorithm, here referred to as the Gauss–Jacobi Combinatorial algorithm, solves the overdetermined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures. Compared to the actual values, the obtained results are more accurate than those obtained from the closed-form solution of a minimano of three known stations.