a control polygon scheme for design of planar ph quintic spline curves
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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
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