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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

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

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Contents Preparation Definition Why How Single knots: Multiple knots : Others

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Preparation B-spline curve: (1) (2) (3)

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Preparation Let n=3, and

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Preparation

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

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Preparation

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

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

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

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

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

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

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

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How( Multiple knots)

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Linear precision property Let are double knots, are collinear. Then the curve lie in is a precision line.

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Linear precision property

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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.

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Local shape modification

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Others Extension to non-uniform knots Closure

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Thank you!

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

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Open PH spline curves

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

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Periodic PH spline curves

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Iteration error

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90 distinct control points

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A “randomized” version

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Iteration error

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

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

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