A control polygon scheme for design of planar PH quintic s

pline curves

Francesca Pelosi Maria Lucia Sampoli

Rida T. Farouki Carla Manni

Speaker:Ying.Liu

2C

Abstract

Control polygon Knot sequence

Pythagorean-hodograph Cubic B-spline curve

Control polygonKnot sequence

Contents

Preparation Definition Why How

Single knots: Multiple knots:

Others

Preparation

B-spline curve:

(1)

(2)

(3)

0

( ) ( )N

nk k

k

c t p B t

1 11

11 1

1( ) ( ) ( )r r rk k r

k k kk r k k r k

t t tB t B t B t

t t t t

11,00,, ( ) { k kt t t

k otherwisewith B t

Preparation

Let n=3, and 0 1 2 3 1 2 3 4( ) ( )N N N Na t t t t t t t t b

Preparation

Preparation

Closed curve: Control points ： Knots: For given ， Let

overlap and overlap

That’s: k=1…n

11201 ,....., nNnNnN pPPPPP

1,........, Nn tt

11 ,......, nNN tt nn tt 2,.......,

ntt ,.......,0 11 ,....... NnN tt

1 1N k N k n k n kt t t t

1 1N n k N n k k kt t t t

Preparation

Definition

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

)()()( 22'2' ttytx )(t

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),

)()( 2' twtr

)()(2)(' tvtuty )()()(' 22 tvtutx

)()()( 22 tvtut

How( single knots)

The curve interpolates ,…… , and , is the end point

of the curve. , and

Let

0q Mq 0q Mq

)(tri ]1,0[t 1)0( ii qr ii qr )1(

221

21

' ])(2

1)1(2)1)((

2

1[)( tzzttztzztr iiiiii

How( single knots)

Interpolation condition Then (10)

End condition

1

0 1' )( iiii qqqdttr

),,( 11 iiii zzzf

06013133273 111121

221 iiiiiiiiii qzzzzzzzzz

For open end condition

For closed end condition

How( single knots)

Nodal points( ): :

iq

Open PH Spline curves:

Periodic PH Spline curves:

How( single knots)

Starting approximation:

(16) And:

(17) Or:

(18)

' '2 3

1 1( ) ( ( ))2 2i i ir c t t

'1 1 2 3

16 8 ( ( ))

2i i i i iz z z c t t

0 1 2 iz z d 1 2M M fz z d

'1 2 3 4

16 8 ( ( ))

2Mz z z c t t

'1 1 1

16 8 ( ( ))

2M M N Nz z z c t t

How( Multiple knots)

How( Multiple knots)

How( Multiple knots)

How( Multiple knots)

Linear precision property Let are double knots,

are collinear. Then the curve lie in is a precision line.

kt 1kt

3 2 1, , ,k k k kp p p p

1( ), ( )k kc t c t

Linear precision property

Linear precision property

How( Multiple knots)

Local shape modification: Let is to be moved. and are doubl

e knots . Then the modified curve is still a PH splin

e ,and well juncture with others.

1,k k nt t kp

Local shape modification

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:

0.... Np p

0 1 2 3 1 2 3 4( ) ...... ( )N N N Na t t t t t t t t b

3( ) 0,..., 2k kq c t k N ' '3 1( ), ( )i f Nd c t d c t

Open PH spline curves

Open PH spline curves

Open PH spline curves

Periodic PH spline curves Definition:

Control points: Periodic knot sequence, Nodal points: End condition:

3( ) 0,..., 2k kq c t k N

0.... Np p

2 0Nq q

Periodic PH spline curves

Periodic PH spline curves

Iteration error

90 distinct control points

A “randomized” version

Iteration error

End conditions

For open curve: and That is:

(12)

)0('1rd i )1('Mf rd

04)(),( 210100 idzzzzf

04)(),( 2111 fMMMMM dzzzzf

End conditions

For closed curve: That is ： and That is:

(13)

0qqM )0()1( '1

' rrM )0()1( ''1

'' rrM

),,( 101 zzzf M

06013133273 211222

21

2 iMMM qzzzzzzzzz

1 1( , , )M M Mf z z z 2 2 21 1 1 1 1 13 27 3 13 13 60 0M M M M M M Mz z z z z z z z z q