Sharpness in Trajectory Estimation for Planar Four-points Piecewise-Quadratic Interpolation

This paper discusses the problem of fitting non-parametric planar data

$Q_m = \{q_i\}^m_{i=0}$

with four-points piecewise-quadratic interpolant to estimate an unknown convex curve

γ

in Euclidean space

E

2

sampled more-or-less uniformly. The derivation of the interpolant involves non-trivial algebraic and symbolic computations. As it turns out, exclusive symbolic computations with

Wolfram Mathematica 9

are unable to explicitly construct the interpolant in question. The alternative solution involves human and computer interaction. The theoretical asymptotic analysis concerning this interpolation scheme as already demonstrated yields quartic orders of convergence for trajectory estimation. This paper verifies in affirmative the sharpness of the above asymptotics via numerical tests and independently via analytic proof based on symbolic computations. Finally, we prove the necessity of admitting more-or-less uniformity and strict convexity to attain at least quartic order of convergence for trajectory approximation. In case of violating strict convexity of

γ

we propose a corrected interpolant

$\bar{Q}$

which preserves quartic order of convergence.