1. Introduction to Subdivision Surfaces

2. Subdivision Curves and Surfaces 4 Subdivision curves –The basic concepts of subdivision. 4 Subdivision surfaces –Important known methods. –Discussion: subdivision vs. parametric surfaces.

3. Corner Cutting

4. 1 : 3 3 : 1

5. Corner Cutting

11. The control polygon The limit curve A control point

12. The 4-point scheme

14. 1 : 1

15. The 4-point scheme 1 : 8

16. The 4-point scheme

27. The control polygon The limit curve A control point

28. Subdivision curves Non interpolatory subdivision schemes Corner Cutting Interpolatory subdivision schemes The 4-point scheme

29. Basic concepts of Subdivision 4 A subdivision curve is generated by repeatedly applying a subdivision operator to a given polygon (called the control polygon). 4 The central theoretical questions: –Convergence : Given a subdivision operator and a control polygon, does the subdivision process converge? –Smoothness : Does the subdivision process converge to a smooth curve?

30. Subdivision schemes for surfaces 4 A Control net consists of vertices, edges, and faces. 4 In each iteration, the subdivision operator refines the control net, increasing the number of vertices (approximately) by a factor of 4. 4 In the limit the vertices of the control net converge to a limit surface. 4 Every subdivision method has a method to generate the topology of the refined net, and rules to calculate the location of the new vertices.

31. Triangular subdivision Works only for control nets whose faces are triangular. Every face is replaced by 4 new triangular faces. The are two kinds of new vertices: Green vertices are associated with old edges Red vertices are associated with old vertices. Old vertices New vertices

32. Loop’s scheme 33 1 1 1 1 1 1 1 n - the vertex valency A rule for new red verticesA rule for new green vertices Every new vertex is a weighted average of the old vertices. The list of weights is called the subdivision mask or the stencil.

33. The original control net

34. After 1st iteration

35. After 2nd iteration

36. After 3rd iteration

37. The limit surface The limit surfaces of Loop’s subdivision have continuous curvature almost everywhere.

38. The Butterfly scheme This is an interpolatory scheme. The new red vertices inherit the location of the old vertices. The new green vertices are calculated by the following stencil: 8 8 2 2

39. The original control net

40. After 1st iteration

41. After 2nd iteration

42. After 3rd iteration

43. The limit surface The limit surfaces of the Butterfly subdivision are smooth but are nowhere twice differentiable.

44. Quadrilateral subdivision Works for control nets of arbitrary topology. After one iteration, all the faces are quadrilateral. Every face is replaced by quadrilateral faces. The are three kinds of new vertices: Yellowfaces Yellow vertices are associated with old faces Green vertices are associated with old edges Red vertices are associated with old vertices. Old vertices New vertices Old edge Old face

45. Catmull Clark’s scheme 1 1 1 1 1 First, all the yellow vertices are calculated Step 1 1 1 1 1 Then the green vertices are calculated using the values of the yellow vertices Step 2 1 1 1 1 1 1 1 1 1 Finally, the red vertices are calculated using the values of the yellow vertices Step 3 n - the vertex valency 1

46. The original control net

47. After 1st iteration

48. After 2nd iteration

49. After 3rd iteration

50. The limit surface The limit surfaces of Catmull-Clarks’s subdivision have continuous curvature almost everywhere.