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Ribs and Fans of Bézier Curves and Surfaces

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

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Ribs and Fans of Bézier Curves and Surfaces

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  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:

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