# Subdivision of Box-Splines

• Malcolm Sabin
Part of the Mathematics and Visualization book series (MATHVISUAL)

## Abstract

The two original refinement algorithms for defining subdivision surfaces were based on the biquadratic and bicubic tensor-product B-splines. At about the same time the use of box-splines as a more inclusive extension of B-splines to multivariate interpolation and approximation was being developed, and fairly soon a refinement algorithm over triangulations based on a box-spline was published.

It turns out that the box-spline provides an excellent context for the presentation of variation diminishing refinement methods, and so this tutorial uses that context as its centre.

The tutorial starts by a brief recapitulation of B-splines and their refinement by knot-insertion, and then shows how the results are achieved more transparently by the use of box-splines and the generating function notation. This is then extended to the bivariate case and to bivariate irregular grids, where the principal schemes are outlined. Finally we consider issues arising in the implementation of refinement algorithms.

## Keywords

Tensor Product Control Point Subdivision Scheme Spline Curve Subdivision Surface
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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