Cubic Polynomial Curves with a Shape Parameter

Transcription

1 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) Cubic olyomial Curves with a Shape arameter MO GUOLIANG ZHAO YANAN Iformatio ad Computatioal Sciece Zheiag Uiversity City College Huzhou Street Hagzhou 005 EOLE S REUBLIC OF CHINA Abstract: A class of cubic polyomial bledig fuctios with a shape parameter is preseted. It is a extesio of cubic uiform B-splie basis fuctios. iecewise polyomial curves with a shape parameter are costructed from these bledig fuctios. The geerated curves have secod geometric cotiuity for ay fixed shape parameter ad have the same termial properties as the cubic uiform B-splie curves. If the value of the shape parameter is chaged the approachig degree of the curves to their cotrol polygo is adusted accordigly ad the curves are maipulated to approximate the cubic uiform B-splie curve from its both sides. I compariso with the existig results the degree of bledig fuctios is lower ad the domai of the shape parameter is larger i this paper. A ew method usig cubic polyomial curves with a shape parameter is also proposed to solve iterpolatio problem without solvig global systems of equatios. Fially some computig examples of the curve desig are also give. Key-Words: Bledig fuctio Shape parameter GC cotiuous Cubic splie Sigular Itroductio As we all ow polyomial B-splie curves especially quadratic ad cubic B-splie curves are widely used i computer aided geometric desig (CAGD). iecewise cotiuous curves with four cotrol poits for each curve segmet are flexible ad ca be used coveietly []. To improve the shape of a curve or the extet which a curve approaches its cotrol polygo the method of costructig curves by usig tesio parameter was preseted i CAGD [-4]. The weights of ratioal Bézier curves ad ratioal B-splie curves have the fuctio to adust the shapes of curves [5]. The shapes of some other high-order ratioal curves ca also be adusted to meet differet eeds [7]. It is good to provide a more utilizable method of geeratig curves with a shape parameter. A C-B splie curve [8] is oe of the curves with a shape parameter ad its basis fuctios cotai triagle fuctios. The shape parameter is represeted as triagle fuctios but usig triagle fuctios is ot as easy as usig polyomial fuctios. Whe values of the shape parameter are chaged C-B splie curves are away from the cotrol polygo. Quadratic ad cubic trigoometric polyomial curves with a shape parameter were proposed successively [90]. The rage of the shape parameter is the iterval[ ]. Aother method of geeratig a cubic uiform B-splie curve with a shape parameter was itroduced []. The degree of its four bledig fuctios is 4 ad the bledig fuctios are ot basis fuctios. The parameter value ca chage o the iterval[ 8]. The curve geerated is C cotiuous but it ca't approximate its cotrol polygo arbitrarily. The method was also geeralized to th( ) order uiform B-splie curve case. Aalogously to the cubic B-splie curves the purpose of this paper is to preset practical piecewise polyomial curves. The method of costructig a cubic polyomial curve with a shape parameter is obtaied ad the degree of its four bledig fuctios is. The value of its shape parameter ca chage o the iterval[ 8 + ). The curve geerated by this method is GC cotiuous ad it ca approximate its cotrol polygo arbitrarily by icreasig the value of its shape parameter. The curve is a extesio of a cubic uiform B-splie curve. It has some good properties as a cubic uiform B-splie curve does. This paper also presets ew iterpolatio method that ca produce C cotiuous cubic polyomial curves with a shape parameter without solvig global systems of equatios [-4] while providig slacess cotrol capabilities. With the low-degree polyomials ad direct computatio of cotrol vertices the method is computatioally simple ad thus useful for iteractive iterpolatio shape desig ad computer graphics applicatios.

2 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) The paper is orgaized as follows. I Sectio the bledig fuctios are proposed ad the properties are described. I Sectio the cubic polyomial curve with a shape parameter is give. Ope ad closed piecewise cubic polyomial curves are described. The approximatio of the GC cotiuous polyomial curve to a cubic B- splie curve ad to the give cotrol polygo are show i Sectio 4. A ew iterpolatio method is also described i Sectio 4. Some coclusios ca be draw i Sectio 5. Cubic basis fuctios with a shape parameter Defiitio. Supposed t [0] λ R the associated bledig fuctios are defied to be the followig fuctios: φ0(; t λ) = ( t) + λ φ (; t λ) = ( + λ) t ( 4+ λ) t λ + λ φ(; t λ) = ( + λ) t + ( + λ) t + t+ + λ φ(; t λ) = t + λ t [0]. Theorem. The bledig fuctios have the followig properties: φ (; t λ) = φ ( t; λ) φ (; t λ) = φ ( t; λ). () 0 () = 0 φ (; t λ) = t [0] λ 8. () φ ( t; λ) 0( = 0 ) t [0] λ 8. (4) φ0 (; t λ) B0 () t φ(; t λ) 8+ λ 8+ λ 4 B() t = φ(; t λ) + λ 4 8+ λ 8+ λ B () t φ (; t λ) B () t ( ) B = t t ( = 0) are Berstei polyomial basis fuctios. roof: (i) roperty () () ad (4) are straightforward. (ii)roperty () is proved accordig to (4). For ay φ (; t λ ) ca be represeted as covex liear combiatios of B i ()( t i = 0) ad the combiatio coefficiets are o-egative so φ (; t λ )( = 0) are o-egative. The proof is completed. If λ = 0 φ ( t;0)( = 0) t [0] are the basis fuctios of cubic uiform B-splie curve o the iterval [0]. The bledig fuctios are the geeralized basis fuctios of cubic uiform B- splie. Fig. shows the images of the basis fuctios for differet shape parameters. Fig. The image of the basis fuctios with differet shape parameters λ Cubic splie curve with a shape parameter Defiitio. Give cotrol poits i ( i = 0 L ) ( ) a ot vector t0 < t < L < t + ad a parameter λ( λ 4 ) the for ay t [ t t + ] = 4 L the curve segmet of polyomial is defied as follows: t t (; t λ ) φ i i ;. i 0 t t λ C = + = + The piecewise cubic polyomial curve is defied as follows: C(; t λ) = C (; t λ ) t [ t t+ ] = 4 L The curve is a piecewise cubic polyomial curve with a shape parameter defied o the iterval[ t t + ] ad it is geeralized from the cubic uiform B-splie curve. For ay t [ t t+ ) = 4 L the curve segmet lies i the covex hull H of. The

3 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) total curve lies i U = H. The shape of ca be adusted whe the parameter value λ is chaged. For t [ t t+ ) = 4 L C (; t λ ) ca be trasformed ito a Bézier curve with its variable t t parameter u = ad its cotrol poits are t+ t b b b b where b ( λ ) ( 8+ λ ) + 4 ( λ ) ( λ ) b = b + λ b similar properties o the edpoit locatio ad edpoit taget vector. If let = 0 ad + = ad 0L + be the cotrol polygo of the curve the 0 ad are its start ad termial poits respectively. ( 0) ad + λ ( ) are its taget vectors at 0 ad + λ respectively. If let = 0 + = + = ad 0L + + be the cotrol polygo of the curve the the curve is a closed curve. Fig. depicts ope ad closed cubic polyomial curves for λ = 0. So de Castelau corer cuttig algorithm of a Bézier curve ca be applied to the computatio of the curve for speed ad stable computatio. Theorem. The curve has d geometric cotiuity i.e. it is a GC cotiuous curve. roof: Let h = t+ t = 4 L. Straightforward computatio is show as: C ( t + ; λ) = + ( 8+ λ) + + λ C( t+ ; λ) = ( 8 λ) λ C ( ; ) ( ) ( ) t+ λ = h + λ C ( t ; λ) = ( ) ( ) + λ + h C ( t + ; λ) = ( 4+ λ) + ( + λ) ( + λ ) h + λ 4+ λ +. λ + C ( t+ ; λ) = h ( ) ( ) ( ) For = 45 L the followig ca be obtaied C( t + ; λ) = C( t ; λ) h C t + = t ( ; λ) C ( ; λ) h h h C ( t+ ; λ) = C ( t ; λ) + λ C ( t ; ). λ h h Accordig to the proof of Theorem the curve ad the cubic uiform B-splie curve have Fig. Ope ad closed GC cotiuous cubic splie curves with λ = 0 I Fig. the curve gradually approximates its cotrol polygo if the shape parameter λ gradually icreases. The property is

4 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) commo. I fact if t is fixed φ () t ad φ () t 0 are mootoe decreasig with respect to λ so for ay C (; t λ ) the weights of poits ad become smaller if λ becomes larger. The curve segmet C (; t λ ) approximates the lie segmet because the weights of poits icrease. ad Fig. Ope GC cotiuous cubic splie curves iterpolatig edpoits with differet λ 4 Applicatio of cubic splie curve with a shape parameter λ ca be used as a shape parameter i order to costruct curves locatig at differet positios. The rage for the shape parameter i this paper is the iterval [ 8 + ). I Refereces [] the shape parameter is the iterval [ 8]. The curves geerated by the two methods ca approximate their cotrol polygo whe the values of their shape parameter icrease. So our method has more abudat modelig fuctios tha that i Refereces []. It is fatal that the bledig fuctios costructed i Refereces [] are degree 4 ad ot basis fuctios. Fig. ad Fig. 4 show examples of curve desig. Fig.4 Ope ad closed GC cotiuous cubic splie curves with λ = (a) Usig method i Referece [] (b) Usig our method If λ [40] the curve approaches more to the cubic uiform B-splie havig the same cotrol polygo with. If λ [0 + ) the curve approaches more to the cotrol polygo of. The iterpolatig problem is cosidered i the followig. Give a set of data poits { i} R 4 our goal is to costruct a lowdegree iterpolatig polyomial curve without i= 0 solvig ay global system to fid its cotrol vertices ad its global slacess cotrolled. I order to iterpolate the two edpoits we itroduce two auxiliary data poits = 0 ad + = if the cotrol polygo is ope ( 0 ) or = ad + = if the cotrol polygo is closed ( 0 = ). The ot vector is obtaied by accumulatig chord-legth.

5 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) The cubic sigular bledig fuctio [4] is defied to be 9 t t t t t + t+ t+ t 9 tt tt t+ t t+ t S () t = t + t+ t t + t+ 9 t + t t + t+ t t+ ; t+ t t < t = L +. + It is easy to prove that S () t are sigular fuctios satisfyig that S( t) = 0 S( t+ ) = S ( t) = S ( t) = 0 S ( t+ ) = S ( t+ ) = 0. The cubicα -splie curve is defied to be Q (; t λ α ) = ( α) C( t; λ) + ( S( t) ) + S( t) α + ( 4 + λ ) + λ + S () t ( α ) + [ + ( + λ) ( + λ) ] + λ t t t = L +. + Q(; t λ α) = Q(; t λ α ) t t t+ = L + It is clear that for ay bledig factor α the resultig curve Q(; t λ α) is GC cotiuous ad Q( t; λ α) = = L +. So the costructed curve Q (; t λ α) iterpolates the data poits { i} i= 0. For the purpose of practical shape modelig we shall restrict α [0] ad call these curves the stadard α cubic splie curve with a shape parameter λ. If α = 0 the α cubic splie curve is reduced to the cubic splie curve. The bledig factor α ca be used to cotrol the slacess of the α cubic splie curve Q (; t λ α). Note that chagig the bledig factor α does ot affect the cotiuity of Q(; t λ α) ad the iterpolatio features at the data poits. Thus there are two parameters that ca ifluece the shape of the cubic splie curve. Fig.5 ad Fig. describe ope ad closed α cubic splie curves with two differet parameters. Fig.5 Ope curves iterpolatig data poits with differet shape parameters ad bledig factors

6 roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April (pp5-70) Fig. Closed curves iterpolatig data poits with differet shape parameters ad bledig factors 5 Coclusios I this paper a type of cubic piecewise polyomial curve with a shape parameter is preseted. The cubic uiform B-splie curve is its special case ad this cubic curve ca locate differet places o both sides of the cubic uiform B-splie curve. By chagig the value of the shape parameter we ca adust the approachig degree of the curves to their cotrol polygo ad maipulate the curves to approximate the cubic uiform B-splie curves from both sides. The curve is GC cotiuous ad it ca arbitrarily approximate its cotrol polygo by icreasig the value of its shape parameter. It ca also be used to solve a iterpolatio problem. The costructed α cubic splie curve iterpolates the give data poits. Some computig examples show that the method of costructig curves has importat use value. Refereces: [] J. Hosche D. Lasser Fudametals of Computer Aided Geometric Desig Wellesley MA (traslated by L.L. Schumaer) 99. [] B. A. Barsy Computer graphics ad geometric modellig usig Beta-splies Spriger-Verlag Hei-delberg 988. [] J. A. Gregory M. Sarfraz A ratioal cubic splie with tesio Computer Aided Geometric Desig Vol pp.-. [4] B. Joe Multiple-ot ad ratioal cubic Betasplies ACM Trasactios o Graphics Vol.8 No. 989 pp [5] L. iegl W. Tiller The NURBS Boo Spriger New Yor 995. [] T. N. T. Goodma Costructig piecewise ratioal curves with Freet frame cotiuity Computer Aided Geometric Desig Vol pp.5-. [7] B. Joe Quartic Beta-splies ACM Trasactios o Graphics Vol.9 No. 990 pp.0-7. [8] Jiwe Zhag C-curves: A extesio of cubic curves Computer Aided Geometric Desig Vol. No. 99 pp [9] Xuli Ha Quadratic trigoometric polyomial curves with a shape parameter Computer Aided Geometric Desig Vol.9 No.7 00 pp [0] Xuli Ha Cubic trigoometric polyomial curves with a shape parameter Computer Aided Geometric Desig Vol. No. 004 pp [] Xuli Ha Shegu Liu A extesio of the cubic uiform B-splie curves Joural of Computer-Aided Desig & Computer Graphics Vol.5 No.5 00 pp [] B. A. Barsy D.. Greeberg Determiig a set of B-splie cotrol vertices to geerate a iterpolatig surface Computer Graph ad Image rocess Vol.4 No. 980 pp.0-. [] K. F. Loe α -B-splie: a liear sigular bledig B-splie. The Visual Computer Vol. 99 pp.8-5. [4] C.L. Tai G.J. Wag Iterpolatio with slacess ad cotiuity cotrol ad covexitypreservatio usig sigular bledig. Joural of Computatioal & Applied Mathematics Vol.7 No. 004 pp.7-.