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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.
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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
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.
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.
Others • Extension to non-uniform knots • Closure
Open PH spline curves • Definition: • Control points: • Knots points: • Nodal points: • End derivatives:
Periodic PH spline curves • Definition: • Control points: • Periodic knot sequence, • Nodal points: • End condition:
End conditions • For open curve: • and • That is: • (12)
End conditions • For closed curve: • That is: and • That is: • (13)