Total Least-Squares Spiral Curve Fitting

Abstract

A rapidly convergent algorithm for fitting clothoids to measured points is developed and tested. The second-order, reduced Hessian method, broadly applicable to the class of scalable, C2 parametrizations, is orthogonal distance regression with four-parameter similarity transformations. The local parameters, or state variables, are implicitly eliminated, and second-order solutions are rigorously computed in the model parameter space (rank ≤4). The algorithm is further distinguished from earlier works by the inclusion of approximation procedures that yield very good starting values. Additionally, a strong connection between the Helmert transformation and the total least-squares problem is established, and a fixed point method is suggested.