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C1 Interpolation with Cumulative Chord Cubics

Kozera R., Noakes L.

Fundamenta Informaticae

2004

Vol. 61, nr 3,4

285--301

Cumulative chord C1 piecewise-cubics, for unparameterized data from regular curves in Rn, are constructed as follows. In the first step derivatives at given ordered interpolation points are estimated from ordinary (non-C1) cumulative chord piecewise-cubics. Then Hermite interpolation is used to generate a C1 piecewise-cubic interpolant. Theoretical estimates of orders of approximation are established, and their sharpness verified through numerical experiments. Good performance of the interpolant is also confirmed experimentally on sparse data.

Asymptotics for Length and Trajectory from Cumulative Chord Piecewise-Quartics

Kozera R.

Fundamenta Informaticae

2004

Vol. 61, nr 3,4

267--283

We discuss the problem of estimating the trajectory of a regular curve γ:[0,T]→Rn and its length d(γ) from an ordered sample of interpolation points Qm = {γ(t0),γ(t1),...,γ(tm)}, with tabular points ti˘s unknown, coined as interpolation of unparameterized data. The respective convergence orders for estimating γ and d(γ) with cumulative chord piecewise-quartics are established for different types of unparameterized data including e-uniform and more-or-less uniform samplings. The latter extends previous results on cumulative chord piecewise-quadratics and piecewise-cubics. As shown herein, further acceleration on convergence orders with cumulative chord piecewise-quartics is achievable only for special samplings (e.g. for ԑ-uniform samplings). On the other hand, convergence rates for more-or-less uniform samplings coincide with those already established for cumulative chord piecewise-cubics. The results are experimentally confirmed to be sharp for m large and n = 2,3. A good performance of cumulative chord piecewise-quartics extends also to sporadic data (m small) for which our asymptotical analysis does not apply directly.