Ryszard KOZERA CURVE MODELING VIA INTERPOLATION BASED ON MULTIDIMENSIONAL REDUCED DATA A Middle Bird’s cable:...”If in The Rainy Island of Every Day Struggle

you cannot find a bird of paradise just get a wet hen.”... Nikita Khrushchev Curve modeling via interpolation based on multidimensional reduced data Summary In this monograph we consider the problem of modeling curves together with the estimation of their length via various interpolation schemes (i.e. piecewise-polynomials) based on discrete reduced data  ¢¡¤£¦¥¨§�©���§�������������§�¡� � �� � � with � ¡����. The latter term defines an ordered sequence of input points in stripped from the corresponding component of parameters. More precisely, reduced data are obtained by sampling a regular parametric curve (sufficiently smooth) (where) in arbitrary Euclidean space without provision of the corresponding tabular parameters, ��©�£ and ��¡�£��- customarily coined in the literature as interpolation knots. In this work, interpolation schemes based on reduced data are termed as non-parametric. On the other hand, fitting non-reduced data (i.e. the pair ¥

Asymptotics for Length and Trajectory from Cumulative Chord Piecewise-Quartics

We discuss the problem of estimating the trajectory of a regular curve fl: [0, T] ! Rnand its length d(fl) from an ordered sample of interpolation points Qm = {fl(t0), fl(t1),..., fl(tm)},with tabular points t0is unknown, coined as interpolation of unparameterized data. The respec-tive convergence orders for estimating fl and d(fl) with cumulative chord piecewise-quartics areestablished for different types of unparameterized data including "-uniform and more-or-less uni-form samplings. The latter extends previous results on cumulative chord piecewise-quadratics and piecewise-cubics. As shown herein, further acceleration on convergence orders with cumulativechord piecewise-quartics is achievable only for special samplings (e.g. for "-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 confirmedto be sharp for m large and n = 2, 3. A good performance of cumulative chord piecewise-quarticsextends also to sporadic data ( m small) for which our asymptotical analysis does not apply directly