Quasi-elastic cubic splines in [formula omitted

Abstract

Given points P1,P2,…,Pn in Rd (d≥2), we consider the problem of constructing a fair interpolating curve. For d=2, we proposed and analyzed, in Johnson and Johnson (2016), a method which first generates a family of G1 interpolating curves, where each piece is a parametric cubic. An energy functional, that loosely approximates bending energy, is defined on this family and then one seeks a curve with minimal energy. Such optimal curves, called quasi-elastic cubic splines, always exist and are always G1, but often they are both G2 and unique. In the present article we extend the construction and analysis to d≥2 and prove sufficient a priori conditions (on the interpolation points) for G2 regularity and uniqueness of the quasi-elastic cubic spline. These sufficient conditions constitute significant improvements over those obtained in Johnson and Johnson (2016). For example, we show that if the exterior angles of the data polygon do not exceed the threshold anglearccos⁡(1/3)≈70.5∘, then the quasi-elastic cubic spline is G2 and unique. In contrast, the threshold angle obtained for d=2 in Johnson and Johnson (2016) is only ≈30.5∘. As in Johnson and Johnson (2016), we first develop a framework and then apply it to the particular example of quasi-elastic cubic splines. This framework potentially applies to other minimal energy interpolation methods.