Department of Computer Science and Technology

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.

Some figures in this document are best viewed in colour. If you received a black-and-white copy, please consult the online version if necessary.

DOI: 10.48456/tr-773


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.

Full text

PDF (6.1 MB)

BibTeX record

  author =	 {Cashman, Thomas J.},
  title = 	 {{NURBS-compatible subdivision surfaces}},
  year = 	 2010,
  month = 	 mar,
  url = 	 {},
  institution =  {University of Cambridge, Computer Laboratory},
  doi = 	 {10.48456/tr-773},
  number = 	 {UCAM-CL-TR-773}