Optimal Geometric Hermite Interpolation of Curves

Bernstein{B ezier two{point Hermite G 2 interpolants to plane and space curves can be of degree up to 5, depending on the situation. We give a complete characterization for the cases of degree 3 to 5 and prove that rational representations are only required for degree 3. x1. Introduction and Overview We consider recovery of curves from irregularly sampled data. If the curves are to be represented by NURBS, we want to generate representations which are minimal in the following sense: 1. they should have a minimal number of knots, 2. their degree should be as small as possible, 3. polynomial pieces are preferred over rational ones, and 4. the sampling and reconstruction process should be independent of the parametrization. Then evaluation algorithms are fast, and additional knot elimination will not be necessary. Furthermore, using minimal degrees usually helps to preserve shape properties. Within the above setting, this paper continues the presentation 14] given at the Biri conference, and we solve some problems posed there. In particular, the examples of 14] showed that the degree of G 2 piece-wise polynomial or rational curve interpolants must necessarily be at least ve in general, while there are cases that work with degree three (de Boor, HH ollig, and Sabin 1], HH ollig 9]) and degree four (Peters 11]). Here, we focus on the problem of determining the minimal degree that works in each speciic situation, and we give a complete classiication. However, we omit Lagrange interpolation and connne ourselves to two{point Hermite interpolation in IR 2 or IR 3. The presentation will mainly be in geometric terms; it started from a complicated algebraic analysis with 19 diierent cases as provided by C. Sch utt 16]. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.