A certification-driven beam-energy projection algorithm for shape-controlled cubic Hermite interpolation

Abstract

We develop a value-only cubic Hermite interpolation method that retains the bending-energy objective of the natural spline while controlling selected geometric features between data points. The method begins with the natural-spline slopes. Three interval tests detect escape from the endpoint envelope, motion opposite to the endpoint chord and excessive one-sided deviation from the chord. A failed test constrains the endpoint slopes, and a global projection recomputes all slopes by minimizing the same energy under the accumulated constraints. The tests and projection are repeated until no additional constraint is required. The process terminates after at most three activations per interval, with every activated interval satisfying its condition throughout. The hybrid solver for each box-constrained quadratic program combines a block active-set iteration, diagonal equilibration and a fallback with quadratic worst-case work. We prove fine-mesh recovery of the natural spline, mesh-independent equilibrated conditioning, worst-case complexity estimates, an exact beam-to-curve operator norm and decay of slope responses with nodal distance when no other slope changes its constraint status. We also estimate the curve change caused by constraint activation. Numerical experiments confirm recovery under refinement and reductions in the measured geometric departures relative to the natural spline.