Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams

Authors:
Yuanxin Liu;Jack Snoeyink
Affiliations:
UNC Chapel Hill, Chapel Hill, NC;UNC Chapel Hill, Chapel Hill, NC
Venue:
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Year:
2007

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Abstract

A long-standing problem in spline theory has been to generalize classic B-splines to the multivariate setting, and its full solution will have broad impact. We initiate a study of triangulations that generalize the duals of higher order Voronoi diagrams, and show that these can serve as a foundation for a family of multivariate splines that generalize the classic univariate B-splines. This paper focuseson Voronoi diagrams of orders two and three, which produce families of quadratic and cubic bivariate B-splines. We believe that these families are the most general bivariate B-splines to date and supportour belief by demonstrating that a classic quadratic box spline, the Zwart-Powell (ZP) element, is contained in our family. Our work is directly based on that of Neamtu, who established the fascinating connection between splines and higher order Voronoi diagrams.