Exact equations of the nonlinear spline

• Authors: John A. Edwards
• Venue: ACM Transactions on Mathematical Software (TOMS)
• Year: 1992

We define the spline interpolating function, and obtain in directly computable form the elementary set of nonlinear equations describing nonlinear spline curves. Using Newton's and Newton-like methods, we solve typical spline configurations, and hence infer that the procedure will reliably yield precise extremum-energy solutions to nonlinear splines of arbitrary (but presumably reasonable) size and complexity.In order to distinguish between stable and unstable states of spline equilibria, we evaluate the energy change resulting from a perturbation, and we briefly discuss aspects of spline existence and uniqueness in relation to the solved examples. We demonstrate the abrupt transition which occurs at the threshold between spline existence and nonexistence, and conclude that proof of a spline's existence is implicit in the solution set of constants yielded by the method.The procedure may be regarded on the one hand as a precise and efficient research instrument for investigating the properties of true splines and elastica, and on the other as an everyday method for obtaining “the smoothest interpolating curve of all.Contact is always maintained with the physical analogue to the curve, the thin uniform elastic beam, since the four assignable parameters used in each spline interval comprise the necessary and sufficient three angles and one length dimension of the actual physical spline.On an historical note, the method may be seen to offer progress in the search, begun in the late 17th century, for a definitive solution to the elastica problem.