Numerical Algorithms

, Volume 50, Issue 4, pp 381–399 | Cite as

Fast and stable evaluation of box-splines via the BB-form

  • Minho Kim
  • Jörg Peters
Original Paper


To repeatedly evaluate linear combinations of box-splines in a fast and stable way, in particular along knot planes, the box-spline is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bézier-form). We show that the BB-coefficients can be derived and stored as integers plus a rational scale factor and derive a hash table for efficiently accessing the polynomial pieces. This pre-processing, the resulting evaluation algorithm and use in a widely-used ray-tracing package are illustrated for splines based on two trivariate box-splines: the seven-directional box-spline on the Cartesian lattice and the six-directional box-spline on the face-centered cubic lattice.


Box-spline Spline Exact evaluation Rational coefficients Bernstein–Bézier-form Face-centered cubic lattice 


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Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Computer and Information Science and EngineeringUniversity of FloridaGainesvilleUSA

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