• Jakob Andreas BærentzenEmail author
  • Jens Gravesen
  • François Anton
  • Henrik Aanæs


Subdivision curves and surfaces are defined as the limit of a sequence of successive refinements of a control polygon or mesh. There are in many cases a close connection to spline curves with a uniform knot vector and uniform tensor product surfaces. However, subdivision surfaces are useful in slightly different scenarios. Put briefly, subdivision is generally more useful for animation, and splines are more useful for geometric design.

First we study subdivision curves. Curve subdivision is simple to express using matrix multiplication, and we discuss the relation to spline curves and how an eigenanalysis can be used to find points on the limit curve (after infinitely many subdivision steps). Then we present a similar discussion but now for subdivision surfaces. Here the matrix representation is somewhat more difficult but still highly useful for analysis of the schemes. The most advanced theory we present is the characteristic map. It is a tool for analysis of whether subdivision schemes are tangent plane continuous in the limit.

Finally we turn to concrete subdivision schemes and discuss the Loop, Catmull–Clark, modified butterfly, \(\sqrt{3}\), and Doo–Sabin schemes. We end with a brief discussion of some advanced techniques and recent methods: parametrization, factorization, and polar subdivision.


Subdivision Scheme Spline Curve Subdivision Surface Control Polygon Loop Subdivision 
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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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Jakob Andreas Bærentzen
    • 1
    Email author
  • Jens Gravesen
    • 2
  • François Anton
    • 1
  • Henrik Aanæs
    • 1
  1. 1.Department of Informatics and Mathematical ModellingTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark

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