Abstract
An order-k univariate B-spline is a parametric curve defined over a set S of at least \(k+2\) real parameters, called knots. Such a B-spline can be obtained as a linear combination of basic B-splines, each of them being defined over a subset of \(k+2\) consecutive knots of S, called a configuration of S.
In the bivariate setting, knots are pairs of reals and basic B-splines are defined over configurations of \(k+3\) knots. Among these configurations, the Delaunay configurations introduced by Neamtu in 2001 gave rise to the first bivariate B-splines that retain the fundamental properties of univariate B-splines. An order-k Delaunay configuration is characterized by a circle that passes through three knots and contains k knots in its interior.
In order to construct a wider variety of bivariate B-splines satisfying the same fundamental properties, Liu and Snoeyink proposed, in 2007, an algorithm to generate configurations. Even if experimental results indicate that their algorithm generates indeed valid configurations, they only succeeded in proving it up to \(k=3\). Until now, no proof has been given for greater k.
In this paper we first show that, if we replace the circles in Neamtu’s definition by maximal families of convex pseudo-circles, then we obtain configurations that satisfy the same fundamental properties as Delaunay configurations. We then prove that these configurations are precisely the ones generated by the algorithm of Liu and Snoeyink, establishing thereby the validity of their algorithm for all k.
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Schmitt, D. (2019). Bivariate B-Splines from Convex Pseudo-circle Configurations. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_23
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