Subdivision of Box-Splines
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 SurfacePreview
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References
- 1.A. A. Ball and D. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACM Trans. on Graphics 7, 1988, 83–102.MATHCrossRefGoogle Scholar
- 2.E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 1978, 350–355 (reprinted in Seminal Graphics, Wolfe (ed.), SIGGRAPH 1998).CrossRefGoogle Scholar
- 3.G. M. Chaikin. An algorithm for high-speed curve generation. Comp. Graphics and Image Proc. 3, 1974, 346–349.CrossRefGoogle Scholar
- 4.D. W. H. Doo and M. A. Sabin. Behaviour of recursive subdivision surfaces near extraordinary points. Computer-Aided Design 10, 1978, 356–360 (reprinted in Seminal Graphics, Wolfe (ed.), SIGGRAPH 1998).CrossRefGoogle Scholar
- 5.C. de Boor. Cutting corners always works. Comput. Aided Geom. Design 4, 1987, 125–132.MATHGoogle Scholar
- 6.C. de Boor, K. Höllig, and S. Riemenschneider. Box Splines. Springer-Verlag, New York, 1993.MATHCrossRefGoogle Scholar
- 7.G. de Rham. Un peu de mathématique a propos d’une courbe plane. Elemente der Mathematik 2, 1947, 73–76, 89–97.Google Scholar
- 8.N. Dyn. Analysis of convergence and smoothness by the formalism of Laurent polynomials. This volume.Google Scholar
- 9.A. R. Forrest. Notes of Chaikin’s algorithm. CGM74–1, University of East Anglia, 1974.Google Scholar
- 10.C. T. Loop. Smooth subdivision surfaces based on triangles. Master’s Thesis, University of Utah, 1987.Google Scholar
- 11.C. T. Loop. Triangle mesh subdivision with bounded curvature and the convex hull property. To appear in The Visual Computer, 2001.Google Scholar
- 12.H. Prautzsch and G. Umlauf. Improved triangular subdivision schemes. Proc. Computer Graphics International 1998, 626–632.Google Scholar
- 13.E. Quak. Nonuniform B-splines and B-wavelets. This volume.Google Scholar
- 14.R. F. Riesenfeld. On Chaikin’s algorithm. IEEE Comp. Graph. Appl. 4, 1975, 304–310.Google Scholar
- 15.M. A. Sabin. Cubic recursive division with bounded curvature. Curves and Surfaces, P.-J. Laurent, A. Le Méhauté, and L. L. Schumaker (eds.), Academic Press, New York, 1991, 411–414.Google Scholar
- 16.M. A. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. This volume.Google Scholar
- 17.A. Sommerfeld. Eine besonders anschauliche Ableitung des Gaußschen Fehlergesetzes. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstag. Verlag von Johann Ambrosius Barth, 1904, 848–859.Google Scholar
- 18.D. Zorin. Stationary Subdivision and Multiresolution Surface Representations. PhD Thesis, California Institute of Technology, 1998.Google Scholar