Introduction to Subdivision Surfaces

1. Introduction toSubdivision Surfaces

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

3. Corner Cutting

4. Corner Cutting 3 : 1 1 : 3

5. Corner Cutting

6. Corner Cutting

7. Corner Cutting

8. Corner Cutting

9. Corner Cutting

10. Corner Cutting

11. A control point The limit curve The control polygon Corner Cutting

12. The 4-point scheme

13. The 4-point scheme

14. The 4-point scheme 1 : 1 1 : 1

15. The 4-point scheme 1 : 8

16. The 4-point scheme

17. The 4-point scheme

18. The 4-point scheme

19. The 4-point scheme

20. The 4-point scheme

21. The 4-point scheme

22. The 4-point scheme

23. The 4-point scheme

24. The 4-point scheme

25. The 4-point scheme

26. The 4-point scheme

27. The 4-point scheme A control point The limit curve The control polygon

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

29. Basic concepts of Subdivision • A subdivision curve is generated by repeatedly applying a subdivision operator to a given polygon (called the control polygon). • 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 • A Control net consists of vertices, edges, and faces. • In each iteration, the subdivision operator refines the control net, increasing the number of vertices (approximately) by a factor of 4. • In the limit the vertices of the control net converge to a limit surface. • 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. New vertices Old vertices • 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.

32. 1 3 3 1 Loop’s scheme Every new vertex is a weighted average of the old vertices. The list of weights is called the subdivision mask or the stencil. A rule for new red vertices A rule for new green vertices 1 1 1 1 1 n- the vertex valency

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. -1 -1 2 8 8 -1 2 -1 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:

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. Old vertices New vertices Old face Old edge • Every face is replaced by quadrilateral faces. • The are three kinds of new vertices: • Yellow vertices are associated with old faces • Green vertices are associated with old edges • Red vertices are associated with old vertices.

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

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.