1. Subdivision Schemes

2. Center for Graphics and Geometric Computing, Technion What is Subdivision?  Subdivision is a process in which a poly-line/mesh is recursively refined in order to achieve a smooth curve/surface.  Two main groups of schemes: Approximating - original vertices are moved Interpolating – original vertices are unaffected Is the scheme used here interpolating or approximating?

3. Center for Graphics and Geometric Computing, Technion Why Subdivision? Frame from “Geri’s Game” by Pixar

4. Center for Graphics and Geometric Computing, Technion  LOD  Compression  Smoothing Why Subdivision? 424Kb 1Kb 52Kb 13.3Mb

5. Center for Graphics and Geometric Computing, Technion Corner Cutting

6. Center for Graphics and Geometric Computing, Technion Corner Cutting 1 : 3 3 : 1

7. Center for Graphics and Geometric Computing, Technion Corner Cutting

8. Center for Graphics and Geometric Computing, Technion Corner Cutting

9. Center for Graphics and Geometric Computing, Technion Corner Cutting

10. Center for Graphics and Geometric Computing, Technion Corner Cutting – Limit Curve

11. Center for Graphics and Geometric Computing, Technion Corner Cutting The control polygon The limit curve – Quadratic B-Spline Curve A control point

12. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

13. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

14. Center for Graphics and Geometric Computing, Technion 4-Point Scheme 1 : 1

15. Center for Graphics and Geometric Computing, Technion 4-Point Scheme 1 : 8

16. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

17. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

18. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

19. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

20. Center for Graphics and Geometric Computing, Technion 4-Point Scheme

21. Center for Graphics and Geometric Computing, Technion 4-Point Scheme The control polygon The limit curve A control point

22. Center for Graphics and Geometric Computing, Technion Comparison Non interpolatory subdivision schemes Corner Cutting Interpolatory subdivision schemes The 4-point scheme

23. Center for Graphics and Geometric Computing, Technion Theoretical Questions  Given a Subdivision scheme, does it converge for all polygons?  If so, does it converge to a smooth curve? Better?  Does the limit surface have any singular points?  How do we compute the derivative of the limit surface?

24. Center for Graphics and Geometric Computing, Technion Surface subdivision  A surface subdivision scheme starts with a control net (i.e. vertices, edges and faces)  In each iteration, the scheme constructs a refined net, increasing the number of vertices by some factor.  The limit of the control vertices should be a limit surface.  a scheme always consists of 2 main parts: A method to generate the topology of the new net. Rules to determine the geometry of the vertices in the new net.

25. Center for Graphics and Geometric Computing, Technion General Notations  There are 3 types of new control points: Vertex points - vertices that are created in place of an old vertex. Edge points - vertices that are created on an old edge. Face points – vertices that are created inside an old face.  Every scheme has rules on how (if) to create any of the above.  If a scheme does not change old vertices (for example - interpolating), then it is viewed simply as if

26. Center for Graphics and Geometric Computing, Technion Loop’s Subdivision - topology  Based on a triangular mesh  Loop’s scheme does not create face points Old face New face Vertex points Edge points

27. Center for Graphics and Geometric Computing, Technion Loop’s subdivision – stencil  Every new vertex is a weighted average of old ones.  The list of weights is called a Stencil  Is this scheme approximating or interpolating? 33 1 1 1 1 1 1 1 n – vertex point’s valence The rule for vertex pointsThe rule for edge points

28. Center for Graphics and Geometric Computing, Technion Loop - Results

29. Center for Graphics and Geometric Computing, Technion Loop - Results

30. Center for Graphics and Geometric Computing, Technion Loop - Results

31. Center for Graphics and Geometric Computing, Technion Loop - Results

32. Center for Graphics and Geometric Computing, Technion Loop - Results Loop’s scheme results in a limit surface which is of continuity everywhere except for a finite number of singular points, in which it is.   Behavior of the subdivision along edges

33. Center for Graphics and Geometric Computing, Technion Butterfly Scheme  Butterfly is an interpolatory scheme.  Topology is the same as in Loop’s scheme.  Vertex points use the location of the old vertex.  Edge points use the following stencil: 8 8 2 2

34. Center for Graphics and Geometric Computing, Technion Butterfly - results

35. Center for Graphics and Geometric Computing, Technion Butterfly - results

36. Center for Graphics and Geometric Computing, Technion Butterfly - results

37. Center for Graphics and Geometric Computing, Technion Butterfly - results

38. Center for Graphics and Geometric Computing, Technion Butterfly - results The Butterfly Scheme results in a surface which is but is not differentiable twice anywhere.

39. Center for Graphics and Geometric Computing, Technion Catmull-Clark  The mesh is the control net of a tensor product B- Spline surface. The refined mesh is also a control net, and the scheme was devised so that both nets create the same B-Spline surface.  Uses face points, edge points and vertex points.  The construction is incremental – First the face points are calculated, Then using the face points, the edge points are computed. Finally using both face and edge points, we calculate the vertex points.

40. Center for Graphics and Geometric Computing, Technion Catmull-Clark 1 1 1 1 1 First, all the face points are calculated Step 1 1 1 1 1 Then the edge points are calculated using the values of the face points and the original vertices Step 2 Last, the vertex points are calculated using the values of the face and edge points and the original vertex Step 3 2 2 2 1 2 1 2 1 1 n - the vertex valence 1 Face points Edge points Vertex points

41. Center for Graphics and Geometric Computing, Technion Connecting The Dots  After Computing the new points, new edges are formed by: connecting each new face point to the new edge points of the edges defining the old face. Connecting each new vertex point to the new edge points of all old edges incident on the old vertex point. Gone

42. Center for Graphics and Geometric Computing, Technion Catmull-Clark - results

43. Center for Graphics and Geometric Computing, Technion Catmull-Clark - results

44. Center for Graphics and Geometric Computing, Technion Catmull-Clark - results

45. Center for Graphics and Geometric Computing, Technion Catmull-Clark - results

46. Center for Graphics and Geometric Computing, Technion Catmull-Clark - results Catmull-Clark Scheme results in a surface which is almost everywhere

47. Center for Graphics and Geometric Computing, Technion Loop Catmull- Clark Butterfly Comparison

48. Center for Graphics and Geometric Computing, Technion Pros and Cons  Pros A single mesh defines the whole model Simple local rules Easy to implement. Numerical Stability Easy to generate sharp feature with Lod, Compression etc  Cons Evaluating a single point on the surface is hard Not suitable for CAGD Mesh topology has great influence on the over all shape. May become expensive in term of rendering Global subdivision