Curve Fitting with Clothoidal Splines

Abstract

Clothoids, i.e. curves Z(s) in R ² whose curvatures x(s) are linear fitting functions of arclength s, have been used for some time for curve fitting purposes in engineering applications. The first part of the paper deals with some basic interpolation problems for clothoids and studies the existence and uniqueness of their solutions. The second part discusses curve fitting problems for clothoidal splines, i.e. C²-curves, which are composed of finitely many clothoids. An iterative method is described for finding a clothoidal spline Z(s) passing through given points Z _i ϵ R ² . i = 0,1,..., n+1, which minimizes the integral ∫ Z x ( s ) 2 d s . This algorithm is superlinearly convergent and needs only 0(n) operations per iteration. A similar algorithm is given for a related problem of smoothing by clothoidal splines.