1. Subdivision Surfaces Introduction to Computer Graphics CSE 470/598 Arizona State University Dianne Hansford

2. Overview What are subdivision surfaces in a nutshell ? Advantages Chaiken’s algorithm The curves that started it all Classic methods Doo-Sabin and Catmull-Clark Extensions on the concept

3. What is subdivision? Input: polygon or polyhedral mesh Process: repeatedly refine (subdivide) geometry Output: “smooth” curve or surface http://www.multires.caltech.edu/teaching/demos/java/chaikin.htm

4. Advantages Easy to make complex geometry Rendering very efficient Animation tools “easily” developed Pixar’s A Bug’s Life first feature film to use subdivision surfaces. (Toy Story used NURBS.)

5. Disadvantages Precision difficult to specify in general Analysis of smoothness very difficult to determine for a new method No underlying parametrization Evaluation at a particular point difficult

6. Chaiken’s Algorithm Chaiken published in ’74 An algorithm for high speed curve generation a corner cutting method on each edge: ratios 1:3 and 3:1

7. Chaiken’s Algorithm Riesenfeld (Utah) ’75 Realized Chaiken’s algorithm an evaluation method for quadratic B-spline curves (parametric curves) Theoretical foundation sparked more interest in idea. Subdivision surface schemes Doo-Sabin Catmull-Clark

8. Doo-Sabin Input: polyhedral mesh one-level of subdivision many levels of subdivision

9. Doo-Sabin ‘78 Generalization of Chaiken’s idea to biquadratic B-spline surfaces Input: Polyhedral mesh Algorithm: 1) Form points within each face 2) Connect points to form new faces: F-faces, E-faces, V-faces Repeat... Output: polyhedral mesh; mostly 4-sided faces except some F- & V-faces; valence = 4 everywhere

10. Doo-Sabin Repeatedly subdivide... Math analysis will say that a subdivision scheme’s smoothness tends to be the same everywhere but at isolated points. extraordinary points Doo-Sabin: non-four-sided patches become extraordinary points

11. Catmull-Clark Input: polyhedral mesh one-level of subdivision many-levels of subdivision

12. Catmull-Clark ‘78 Input: Polyhedral mesh Algorithm: 1)Form F-vertices: centroid of face’s vertices 2)Form E-points: combo of edge vertices and F-points 3)Form V-points: average of edge midpoints 4)Form new faces (F-E-V-E) Repeat.... Output: mesh with all 4-sided faces but valence not = 4 Generalization of Chaiken’s idea to bicubic B-spline surfaces

13. CC - Extraordinary Valence not = 4 1) Input mesh had valence not = 4 2) Face with n>4 sides Creates extraordinary vertex (in limit) (Remember: smoothness less there)

14. Let’s compare D-S C-C

15. Convex Combos Note: D-S & C-C use convex combinations ! (Weighting of each point in [0,1]) Guarantees the following properties: new points in convex hull of old local control affinely invariant (All schemes use barycentric combinations) See references at end for exact equations

16. Data Structures  Each scheme demands a slightly different structure to be most efficient  Basic structure for mesh must exist plus more info  Schemes tend to have bias – faces, vertices, edges.... as foundation of method  Lots of room for creativity!

17. Extensions Many schemes have been developed since.... more control (notice sharp edges) See NYU reference for variety of schemes interpolation (butterfly scheme) Pixar: tailored for animation

18. References Ken Joy’s class notes http://graphics.cs.ucdavis.edu http://graphics.cs.ucdavis.edu Gerald Farin & DCH The Essentials of CAGD, AK Peters http://eros.cagd.eas.asu.edu/~farin/essbook/essbook.html http://eros.cagd.eas.asu.edu/~farin/essbook/essbook.html Joe Warren & Heinrik Weimer www.subdivision.org www.subdivision.org NYU Media Lab http://www.mrl.nyu.edu/projects/subdivision/ http://www.mrl.nyu.edu/projects/subdivision/ CGW article http://cgw.pennnet.com/Articles/Article_Display.cfm?Section=Articles&Subsection=Display& ARTICLE_ID=196304 http://cgw.pennnet.com/Articles/Article_Display.cfm?Section=Articles&Subsection=Display& ARTICLE_ID=196304