Resultant optimization of the three-dimensional intersection problem

Abstract

Resultant approach is here employed to optimize the dimensionless space angles to solve in a closed form the over-determined three-dimensional intersection problem. The advantages of the resultant optimization approach are the non-requirement of the approximate initial starting values, non iterative and does not rely on linearization during its operation, save for the nonlinear variance-covariance/error propagation to generate the weight matrix. Resultant method, a branch of abstract algebra, is employed to compute the combinatorial scatters, which are then optimized to offer a closed form solution. Using the test network Stuttgart Central as an example, it is demonstrated that the resultant optimization approach can be applied as an alternative approach to conventional methods such as least squares for point positioning within the over-determined intersection framework, especially when the approximate starting values for linearization and iterative approaches are not known as may happen in Photogrammetry, Machine Vision or in Robotics.