# 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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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

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

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

Preparation Let n=3, and

Preparation

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

Preparation

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

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

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

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

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

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

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

How( Multiple knots)

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

Linear precision property

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.

Local shape modification

Others Extension to non-uniform knots Closure

Thank you!

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

Open PH spline curves

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

Periodic PH spline curves

Iteration error

90 distinct control points

A “randomized” version

Iteration error

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

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

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