Computer Laboratory

Technical reports

NURBS-compatible subdivision surfaces

Thomas J. Cashman

March 2010, 99 pages

This technical report is based on a dissertation submitted January 2010 by the author for the degree of Doctor of Philosophy to the University of Cambridge, Queens’ College.

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Abstract

Two main technologies are available to design and represent freeform surfaces: Non-Uniform Rational B-Splines (NURBS) and subdivision surfaces. Both representations are built on uniform B-splines, but they extend this foundation in incompatible ways, and different industries have therefore established a preference for one representation over the other. NURBS are the dominant standard for Computer-Aided Design, while subdivision surfaces are popular for applications in animation and entertainment. However there are benefits of subdivision surfaces (arbitrary topology) which would be useful within Computer-Aided Design, and features of NURBS (arbitrary degree and non-uniform parametrisations) which would make good additions to current subdivision surfaces.

I present NURBS-compatible subdivision surfaces, which combine topological freedom with the ability to represent any existing NURBS surface exactly. Subdivision schemes that extend either non-uniform or general-degree B-spline surfaces have appeared before, but this dissertation presents the first surfaces able to handle both challenges simultaneously. To achieve this I develop a novel factorisation of knot insertion rules for non-uniform, general-degree B-splines.

Many subdivision surfaces have poor second-order behaviour near singularities. I show that it is possible to bound the curvatures of the general-degree subdivision surfaces created using my factorisation. Bounded-curvature surfaces have previously been created by ‘tuning’ uniform low-degree subdivision schemes; this dissertation shows that general-degree schemes can be tuned in a similar way. As a result, I present the first general-degree subdivision schemes with bounded curvature at singularities.

Previous subdivision schemes, both uniform and non-uniform, have inserted knots indiscriminately, but the factorised knot insertion algorithm I describe in this dissertation grants the flexibility to insert knots selectively. I exploit this flexibility to preserve convexity in highly non-uniform configurations, and to create locally uniform regions in place of non-uniform knot intervals. When coupled with bounded-curvature modifications, these techniques give the first non-uniform subdivision schemes with bounded curvature.

I conclude by combining these results to present NURBS-compatible subdivision surfaces: arbitrary-topology, non-uniform and general-degree surfaces which guarantee high-quality second-order surface properties.

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BibTeX record

@TechReport{UCAM-CL-TR-773,
  author =	 {Cashman, Thomas J.},
  title = 	 {{NURBS-compatible subdivision surfaces}},
  year = 	 2010,
  month = 	 mar,
  url = 	 {http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-773.pdf},
  institution =  {University of Cambridge, Computer Laboratory},
  number = 	 {UCAM-CL-TR-773}
}