1. Ribs and Fans Ribs and Fans of Bézier Curves and Surfaces Reporter: Dongmei Zhang 2007.11.21

2. Papers (Joo-Haeng Lee and Hyungjun Park) Ribs and Fans of Bézier Curves and Surfaces Computer-Aided Design & Applications 2005 Geometric Properties of Ribs and Fans of a Bézier Curve CKJC 2006, Hangzhou A Note on Morphological Development and Transformation of Bézier Curves based on Ribs and Fans SPM 2007, Beijing

3. Definition and Decomposition Ribs and Fans of Bézier Curves and Surfaces Computer-Aided Design & Applications 2005

4. Ribs and Fans curvesurface

5. Ribs of a Bézier Curve A Bézier curve: Rib control points: Rib (a Bézier curve of degree k):

6. Examples (cubic Bézier curve) A cubic Bézier curve Control points of ribs Rib curves

7. Especial property

8. Fans of a Bézier Curve A Bézier curve: Fan control vectors: Fan (“a Bézier curve” of degree k):

9. Examples (cubic Bézier curve) Control vectors Fans

10. Decomposition Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.

11. Proof (mathematical induction) Base step: n=2

12. Proof Induction hypothesis (n=k): n=k+1 :

13. Proof

15. Decomposition Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.

16. Decomposition Corollary: A Bézier curve of degree n can be decomposed into a single rib of degree l and a sequence of n-l fans of degrees from n-2 to l-1.

17. Surface case

18. Composite transformation

19. Rib and its control points

20. Fan and its control vectors

21. Decomposition Theorem: A Bézier surface of degree (m,n) can be decomposed into a rib of degree (m-1,n-1) and three fans.

22. Proof 固定 v 在 u 方向上 固定 u 在 v 方向上

23. Decomposition Corollary: A Bézier surface of degree (m,n) can be decomposed into a single rib of degree (m- k,n-k) and a sequence of k composite fans.

24. Example(bi-cubic Bézier surface)

26. Examples (Bézier curve of degree 9) d

27. Examples (Bézier curve of degree 10)

28. Geometric Properties of Ribs and Fans of a Bézier Curve CKJC 2006, Hangzhou

29. Composite fans Rib-invariant deformation

30. Composite fans Property 1: A Bézier curve of degree n can be composed into a straight line segment and a composite fan of degree n-2. degree elevation

31. Composite fans Property 2: A straight line segment and a composite fan of degree n-2 can build a unique Bézier curve of degree n.

32. Proof degree elevation

33. Rib-invariant Deformation Property 3: For a given Bézier curve of degree n, we can modify up to n-d control points while preserving a rib of degree d. Moreover, if we specify n-d control points explicitly, we can determine the unknown d-1 control points uniquely.

34. Proof Initial Bézier curve: Rib of degree d: New Bézier curve:

35. Example (quartic Bézier curve)

36. Example (curve of degree 9)

37. Applications A Note on Morphological Development and Transformation of Bézier Curves based on Ribs and Fans SPM 2007, Beijing

38. Morphological development To find a sequence of shapes that believed to represent a pattern of growth.

39. Morphological transformation To find a sequence curves that represents the pattern from one curve to another.

40. Morphological development Current shape: Initial shape (simple, minimum features): Developmental pattern:

41. DCF (development by composite fan) Linear development:

42. DCF (development by composite fan)

43. DFL (development by fan lines) Utilize each rib:

44. DFL (development by fan lines)

46. DSC (development by spline curves) path: a smooth curve.

47. DSC (development by spline curves)

48. Comparision

49. Morphological transformation Three methods (TLI,TCE,TDE). TLI (Transformation by linear interpolation). correspondence: index of control points.

50. TCE (by cubic blending and extrapolation) Two Bézier curves: Lower ribs:

51. TDE (by development and extrapolation)

52. Comparision

53. Summary Definition: Ribs and Fans Decomposition. Bézier curve = rib + scaled fans. Property. Rib + composite fan= Bézier curve=rib. Rib-invariant deformation. Morphological applications Development and transformation.

54. Thanks