Geometrical Differentiation and High—Accuracy Curve Interpolation

Abstract

Let f be a smooth curve in ℝ^d, parametrized by arclength. If a large sample of data points p _i = f(t _i) at unknown parameter values t _i < t_i+1 is given, one can use local n-th degree polynomial interpolation at parameters s _i = ∥P _i — Pi _ℓ∥sgn(i — ℓ) of data points P _i around a fixed point P _i to calculate approximations to the derivatives with accuracy O (h ^n+1-J), where h:= max(t_i — t_i-1) and 0 ≤j ≤k — 1 ≤ n. Using these as data for properly parametrized Hermite interpolation problems for polynomials of degree ≤ 2k — 1 ≤ n between successive data points, one can construct GC ^k-1 interpolants of f with accuracy O(h ^2k).