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A control polygon scheme for design of planar PH quintic spline curves

A control polygon scheme for design of planar PH quintic spline curves. Francesca Pelosi Maria Lucia Sampoli Rida T. Farouki Carla Manni Speaker:Ying.Liu. Abstract. Control polygon Knot sequence Pythagorean-hodograph Cubic B-spline curve. Control polygon Knot sequence.

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A control polygon scheme for design of planar PH quintic spline curves

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  1. A control polygon scheme for design of planar PH quintic spline curves Francesca Pelosi Maria Lucia Sampoli Rida T. Farouki Carla Manni Speaker:Ying.Liu

  2. Abstract • Control polygon • Knot sequence • Pythagorean-hodograph • Cubic B-spline curve Control polygon Knot sequence

  3. Contents • Preparation • Definition • Why • How • Single knots: • Multiple knots: • Others

  4. Preparation • B-spline curve: • (1) • (2) • (3)

  5. Preparation • Let n=3, • and

  6. Preparation

  7. Preparation • Closed curve: • Control points： • Knots: For given ，Let overlap and overlap • That’s: k=1…n

  8. Preparation

  9. Definition • Polynomial curve r (t)=(x (t) ,y (t)) ,satisfies for some polynomial

  10. Why • Rational offset curves • Exact arc length • Well-suited real-time CNC interpolator algorithm

  11. How( Single knots) • Let r (t)=x (t) +i y (t), • w (t)=u (t)+ i v (t),

  12. How( single knots) • The curve interpolates ,…… , and , is the end point of the curve. • , and • Let

  13. How( single knots) • Interpolation condition • Then • (10) • End condition For open end condition For closed end condition

  14. How( single knots) • Nodal points( ): • : Open PH Spline curves: Periodic PH Spline curves:

  15. How( single knots) • Starting approximation: • (16) • And: • (17) • Or: • (18)

  16. How( Multiple knots)

  17. How( Multiple knots)

  18. How( Multiple knots)

  19. How( Multiple knots) • Linear precision property • Let are double knots, are collinear. Then the curve lie in is a precision line.

  20. Linear precision property

  21. Linear precision property

  22. How( Multiple knots) • Local shape modification: • Let is to be moved. and are double knots . • Then the modified curve is still a PH spline ,and well juncture with others.

  23. Local shape modification

  24. Local shape modification

  25. Others • Extension to non-uniform knots • Closure

  26. Thank you!

  27. Open PH spline curves • Definition: • Control points: • Knots points: • Nodal points: • End derivatives:

  28. Open PH spline curves

  29. Open PH spline curves

  30. Open PH spline curves

  31. Periodic PH spline curves • Definition: • Control points: • Periodic knot sequence, • Nodal points: • End condition:

  32. Periodic PH spline curves

  33. Periodic PH spline curves

  34. Iteration error

  35. 90 distinct control points

  36. A “randomized” version

  37. Iteration error

  38. End conditions • For open curve: • and • That is: • (12)

  39. End conditions • For closed curve: • That is： and • That is: • (13)

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