Ribs and Fans of Bézier Curves and Surfaces

1. Ribs and Fansof 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 curve surface

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 Rib curves Control points of ribs

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) Fans Control vectors

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

14. 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: ABézier surface of degree (m,n) can be decomposed into arib 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)

25. Example(bi-cubic Bézier surface)

26. Examples (Bézier curve of degree 9) • d

27. Examples (Bézier curve of degree 10)

28. Geometric Properties 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 (developmentby composite fan) • Linear development:

42. DCF(developmentby composite fan)

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

44. DFL(development by fan lines)

45. 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: