Subdivision of Bezier curves

1. Subdivision of Bezier curves RaedaNaamnieh

2. Outline

3. motivation

4. Definitions • Definition 5.7 For , the functions for where n is any nonnegative integer, are called the generalized Bernstein blending functions.

5. Definitions • Definition 16.11 We call the Bezier curve with control points on the interval .

6. The Bezier Curve Subdivision Theorem • For defined as above then where

7. The Bezier Curve Subdivision Theorem

8. The Bezier Curve Subdivision Theorem

9. Outline

10. Restricted Proof for Bezier Subdivision • Lemma 16.22

11. Restricted Proof for Bezier Subdivision • Proof:

12. Restricted Proof for Bezier Subdivision • Proof:

13. Restricted Proof for Bezier Subdivision • Proof for Bezier Subdivision: induction on n, and for arbitrary c, a<c<b. If n=1

14. Restricted Proof for Bezier Subdivision • Proof for Bezier Subdivision: Now, assume the theorem holds for all

15. Restricted Proof for Bezier Subdivision • Proof for Bezier Subdivision: Now using the results from

16. Restricted Proof for Bezier Subdivision • Proof for Bezier Subdivision: -The second part of the proof is almost identical, hence left as exercise

17. Outline

18. Convergence of Refinement Strategies Subdivision at the midpoint

19. Convergence of Refinement Strategies

20. Convergence of Refinement Strategies

21. Convergence of Refinement Strategies

22. Convergence of Refinement Strategies

23. Convergence of Refinement Strategies

24. Convergence of Refinement Strategies • Bezier polygon defined on . • the piecewise linear function given by the original polygon. • the piecewise linear function formed with vertices defined by concatenating together the control polygons for the two subdivided curves and at the midpoint. • It has 2n+1 distinct points.

25. Convergence of Refinement Strategies • is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

26. Convergence of Refinement Strategies • is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

27. Convergence of Refinement Strategies • is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

28. Convergence of Refinement Strategies • The subdivided Bezier curve at level is over the interval: and has vertices: for • We shall write has distinct points which define it.

29. Convergence of Refinement Strategies Theorem 16.17: That is, the polyline consisting of the union of all the sub polygons converges to the Bezier curve.

30. Convergence of Refinement Strategies Lemma 16.18: If is a Bezier curve, define . If Are defined by the rule in Theorem 16.12, then for

31. Convergence of Refinement Strategies Proof: By induction on the superscript, for ,

32. Convergence of Refinement Strategies Proof: Now, suppose that the conclusion has been shown for superscripts up to .Then,

33. Convergence of Refinement Strategies Lemma 16.19: Any two consecutive vertices of are no farther apart than ,where is independent of . That is, if and are two consecutive vertices of Then .

34. Convergence of Refinement Strategies Proof: Induction on , Let First consider and

35. Convergence of Refinement Strategies Proof: Let where

36. Convergence of Refinement Strategies Proof: Now, suppose

37. Convergence of Refinement Strategies Proof: Assume for . Now we show it is true for . The vertices in are defined by subdividing the Bezier polygons in . We see that are formed by subdividing the Bezier curve with control polygon where respectively.

38. Convergence of Refinement Strategies Proof: We shall prove the results for Let us fix And call By the subdivision Theorem 16.12

39. Convergence of Refinement Strategies Proof: Since this is proved for all the conclusion of the lemma holds for all

40. Convergence of Refinement Strategies Proof for convergence theorem: The subdivision theorem showed that over each subinterval , the Bezier curve resulting from the appropriate sub collection of is identical to the original We denote this by .

41. Convergence of Refinement Strategies Proof for convergence theorem: Any arbitrary value in the original interval is then contained in an infinite sequence of intervals, for which

42. Convergence of Refinement Strategies Proof for convergence theorem: Hence, the curve value, lies within the convex hull of the vertices of which correspond to the Bezier polygon over , for each .

43. Convergence of Refinement Strategies Proof for convergence theorem: Since the spacial extent of the convex hull of each Bezier polygon over , all and , gets smaller and converges to zero.

44. Convergence of Refinement Strategies Proof for convergence theorem: Consider the subsequence of polygons corresponding to the intervals containing . is contained in all of them, for all Further, if any other curve point were contained in all of them, say , then would be in

45. Convergence of Refinement Strategies Proof for convergence theorem: Since is the only point in that intersection, is the only point in the intersection of the convex hull of the Bezier polygons of these selected subintervals. The polygonal approximation converges.

46. Summary

47. Appendix Geometric Modeling with Splines Chapter 16 Elaine Cohen Richard F. Riesenfeld GershonElber

48. Q&A