Course Title

Subdivision for Modeling and Animation

sample images

Course Materials

Course notes (30MB PDF).


This course provides an introduction to Subdivision, a technique to generate smooth curves and surfaces, which extends classical spline modeling approaches. The course will cover the basic ideas of subdivision as well as the particulars of a number of different subdivision algorithms; we will present the most recent contributions to the area in a form accessible to a wide audience. The emphasis will be on practical issues in using subdivision for geometric modeling and animation.

Course Presenters


Denis Zorin
Media Research Laboratory
715 Broadway, Rm 1201
New York University
New York, NY 10003
Peter Schröder
Caltech Multi-Res Modeling Group
Computer Science Dept. 256-80
California Institute of Technology
Pasadena, CA 91125


Tony DeRose
Studio Tools Group
Pixar Animation Studios
1001 West Cutting Blvd.
Richmond, CA 94804
Leif Kobbelt
Computer Graphics Group
Max-Planck-Institute for Computer Sciences
Im Stadtwald
66123 Saarbr ücken, Germany
Adi Levin
School of Mathematics
Tel-Aviv University
69978 Tel-Aviv Israel
Wim Sweldens
Bell Laboratories, Lucent Technologies
600 Moutain Avenue
Murray Hill, NJ 07974

Expanded Statement

Subdivision is an algorithmic technique to generate smooth surfaces as a sequence of successively refined polyhedral meshes. Its origins go back to 1978 when Catmull and Clark, and Doo and Sabin first proposed to generalize spline-patch methods to meshes of arbitrary topology. Subdivision algorithms are exceptionally simple, work for arbitrary control meshes and produce globally smooth surfaces. Special choices of subdivision rules allow for the introduction of features into a surface in a simple way. Subdivision-based representations of complex geometry can be manipulated and rendered very efficiently, which makes subdivision a highly suitable tool for interactive animation and modeling systems.

This course will cover the basic ideas of subdivision and a variety of different subdivision schemes detailing their properties, suitability for particular applications, and compare their relative merits. Strong emphasis will be placed on practical issues. At the end of the course participants will be well prepared to implement the basic techniques as well as delve into the research literature on the subject.


The course will be self contained and does not assume prior knowledge of subdivision. Prior exposure to parametric representations of curves and surfaces and associated algorithms is required. Beyond that only basic linear algebra and introductory calculus are required.

Topics Beyond the Prerequisites

The course will teach the basic ideas of subdivision for curves and surfaces. We will explain a number of different subdivision methods for surfaces and briefly discuss the theoretical results (without proofs) useful in practice. The course will cover a variety of subdivision-based algorithms for modeling and animation, including adaptive evaluation, level-of-detail rendering and manipulation. A number of specific applications will be presented.

Course Syllabus


The morning section will focus on the foundations of subdivision, starting with subdivision curves and moving on to surfaces. We will review and compare a number of different schemes and discuss the relation between subdivision and splines. The emphasis will be on properties of subdivision most relevant for applications.
  • Introduction and overview (Schröder); 15 min.
    • Course outline and schedule
    • High-level introduction to the basic ideas of subdivision
    • Quick overview of application examples
  • Foundations I: Basic Ideas (Schröder) 60 min
    • Constructing smooth curves through subdivision; 10 min.
      examples: b-spline knot insertion and interpolating subdivision
    • Subdivision for surfaces; 10 min.
      an example of a subdivision scheme: Loop
    • Properties of subdivision schemes: smoothness, locality, hierarchical structure; 10 min.
    • How splines are related to subdivision; 10 min.
    • Advantages of subdivision: arbitrary topology, efficiency, controllable surface features such as creases and cusps; 10 min.
    • Questions and answers; 10 min.
  • Foundations II: Subdivision Schemes for Surfaces (Zorin), 90 min.
    • Overview of subdivision for surfaces, 15 min.
    • Subdivision matrices for surface schemes; computing tangents and limit positions 15 min.
    • Subdivision rules for special surface features; boundaries and creases; 10 min.
    • Classic schemes, their definition, basic properties and comparison, 25 min.
      1. Catmull-Clark
      2. Doo-Sabin
      3. Loop
      4. Butterfly
      5. Midedge
      6. Kobbelt
    • Explicit Evaluation of subdivision surfaces. 15 min.
    • Questions and answers; 10 min.


The afternoon session will focus on applications of subdivision and the algorithmic issues practitioners need to address to build efficient, well behaving systems for modeling and animation with subdivision surfaces. Each presentation will be 30 min. long, with 10 min. allocated for questions and discussion.
  • Applications and Algorithms:
    • Implementing Subdivision and Multiresolution Surfaces, 40 min. Subdivision can model smooth surfaces, but in many applications one is interested in surfaces which carry details at many levels of resolution. Multiresolution mesh editing extends subdivision by including detail offsets at every level of subdivision, unifying patch based editing with the flexibility of high resolution polyhedral meshes. In this part, we will focus on implementation concerns common for subdivision and multiresolution surfaces based on subdivision. (Zorin)
    • Combined Subdivision Schemes, 40 min. The speaker will present a class of subdivision schemes called "Combined Subdivision Schemes". These are subdivision schemes whose limit surfaces can satisfy prescribed boundary conditions. Every combined subdivision scheme consists of an ordinary subdivision scheme that operates in the interior of the mesh, and special rules that operate near tagged edges of the mesh and take into consideration the given boundary conditions. The limit surfaces are smooth and they satisfy the boundary conditions. The speaker will present examples of combined subdivision schemes, and discuss their applications. (Levin)
    • Parameterization, remeshing, and compression using subdivision, 40 min. Subdivision methods typically use a simple mesh refinement procedure such as triangle or square quadrisection. Iterating this refinement step starting starting from a coarse arbitrary connectivity control mesh generates so-called semi-regular meshes. However meshes coming from scanning devices are fully irregular and do not have semi-regular connectivity. In order to use multiresolution and subdivision based algorithms for such meshes they first need to be remeshed onto semi-regular connectivity. In this talk we show how to use mesh simplification to build a smooth parameterization of dense irregular connectivity meshes and to convert them to semi-regular connectivity. Our method supports both fully automatic operation as well as and user defined point and edge constraints. We also show how semi-regular meshes can be compressed using a wavelet and zero-tree based algorithm. (Sweldens)
    • A Variational Approach to Subdivision, 40 min. Surfaces generated using subdivision have certain orders of continuity. However, it is well known from geometric modeling that high quality surfaces often require additional optimization (fairing). In the variational approach to subdivision, refined meshes are not prescribed by static rules, but are chosen so as to minimize some energy functional. The approach combines the advantages of subdivision (arbitrary topology) with those of variational design (high quality surfaces). This section will describe the theory of variational subdivision and highly efficient algorithms to construct fair surfaces. (Kobbelt)
    • Subdivision Surfaces in the Making of Geri's Game, A Bug's Life, and Toy Story 2, 40 min. Geri's Game is a 3.5 minute computer animated film that Pixar completed in 1997. The film marks the first time that Pixar has used subdivision surfaces in a production. In fact, subdivision surfaces were used to model virtually everything that moves. Subdivision surfaces went on to play a major role the feature films 'A Bug's Life' and 'Toy Story 2' from Disney/Pixar. This section will describe what led Pixar to use subdivision surfaces, discuss several issues that were encountered along the way, and present several of the solutions that were developed. (DeRose)
  • Summary and Wrapup: (all speakers)