Physical Splines

Abstract

Let p₁,…, p _n ∈ R³. A flexible springy wire of length h pinned at the end points p₁ and p _n , and passing through the points p₂, … , p_n−1 with or without specified directions is physically determined to have one of several stable shapes, each corresponding to a locally minimal energy value. We may seek these minimal energy shapes; such a curve will be called a physical spline curve. Although a cubic spline may be a good approximation to a physical spline, this is often not the case, and moreover the length constraint applied to a cubic spline segment is difficult to honor. It is important in various engineering applications to be able to compute the exact physical spline curve of a given length that satisfies given boundary conditions.