Least Squares Comparison of Two Rigorous Math Models Used to Fit a Triaxial Ellipsoid to a Cluster of Points

Abstract

The least-squares (LS) method is universally embraced to obtain approximate unique solutions and their statistics from overdetermined scientific experiments where the number of observations amply exceeds the number of unknowns. The so-called mathematical model is the functional relationship marquee controlling the quality of the solution. A correct math model that faithfully recreates the physical properties of nature is essential for obtaining meaningful results. In this investigation, the LS outcomes using two totally different and accurate math models are compared and, amazingly, the values of the parameters (aka unknowns) computed are exactly the same as the original published values. This new exercise corroborates that, as intuitively anticipated, both solutions agree—although, remarkably, much closer than expected. In the process of this revamped research, novel innovative notation and ideas related to taking partial derivatives required to generate the design matrices are explained and, step-by-step, implemented. The derivations are presented introducing a user-friendly didactic path aimed to help readers (primarily students) working on the physical sciences and engineering to comprehend and master the entire methodology. A Matlab simple function to fit a triaxial ellipsoid to a cluster of points was coded based on the formulation developed for this study.