Geometric Hermite interpolation with maximal order and smoothness

Abstract

We conjecture that, under suitable assumptions, splines of degree ⩽ n can interpolate points on a smooth curve in R m with order of contact k − 1 = n − 1 + [ (n − 1) (m − 1) ] at every nth knot. Moreover, this Geometric Hermite Interpolant (GHI) has the optimal approximation order k + 1. We give a proof of this conjecture for planar quadratic spline curves and describe a simple construction of curvature continuous quadratic splines from control polygons.