American International Journal of Research in Science, Technology, Engineering & Mathematics

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1 Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs Avalable onlne at ISSN (Prnt): 8-9, ISSN (Onlne): 8-8, ISSN (CD-ROM): 8-69 AIJRSTEM s a refereed, ndexed, peer-reveed, multdscplnary and open access journal publshed by Internatonal Assocaton of Scentfc Innovaton and Research (IASIR), USA (An Assocaton Unfyng the Scences, Engneerng, and Appled Research) Constructon of Ratonal Quartc Trgonometrc Bézer Curves Mrdula Dube & Urvash Mshra Professor, Department of Mathematcs and Computer Scence, R.D. Unversty, Jabalpur, Madhya Pradesh, Inda Assstant Professor, Department of Mathematcs, Mata Gujr Mahla Mahavdyalaya, Jabalpur,Madhya Pradesh, Inda Abstract: In ths paper a ne knd of ratonal quartc trgonometrc Bézer curve s defned. These curves not only have most propertes of the quartc Bézer curves th the Bernsten bass n the polynomal space, but also have other propertes for shape modellng. The shape parameter provdes freedom for desgn and shape control of the curve. Here e gve a shape preservng nterpolaton condton based on shape parameter. Thus e can easly construct smooth curves of any shape. The shape of the curve can be adjusted by alterng the values of shape parameter hle the control polygon s kept unchanged. These curves can be used as an effcent model for geometrc desgn n the felds of CAGD. Keyords: Quartc Trgonometrc Bézer bass functon, Quartc ratonal Trgonometrc Bézer curve, Quartc ratonal Trgonometrc Bézer surface, Shape parameter. I. INTRODUCTION Interpolaton of an order sequence of ponts s one of the most dely use of methods n practcal curve modelng. Hence, there ere a vast number of papers, books and chapters dealng th ths topc. Desgners generally prefer splne curves, here most of the methods ork globally. Splne Interpolaton s a sgnfcant tool n computer graphcs, computer aded geometrc desgn (CAGD) and engneerng as ell. The fttng of a splne curve to a set of data ponts has applcaton n computer-aded-desgn (CAD), CAM, computer graphc system, robot path and trajectory plannng. B-Splne curves and ther ratonal generalzaton play a central role and are dely used n computer aded desgn today. These methods are excellent tool n desgn system to create ne objects, but the modfcaton and shape control of the exstng objects are also essental. The Bezer-Curves and surfaces form a basc tool for constructng free form curves and surfaces. The study of curves and surfaces s key element n CAGD & CAD/CAM that has been around for qute some tme. The method of CAGD has arsen from the need of effcent computer representaton of practcal curves and surfaces used n engneerng desgn. Therefore, on ths feld t s desrable to generate a convexty preservng nterpolatng curves and surfaces based on gven data. Convexty s a substantal shape characterstc of the data. The sgnfcance of the convexty preservng nterpolaton problem n ndustry can not be dened. A number of examples can be quoted n ths regard, lke the modelng of cars n automoble ndustry, aero plane and shp desgn. Desgnng ell shaped smooth curve and surfaces also arse n manufacturng the TV-Screens. In the surface desgnng sense e say that the screens must preserve the convexty. Shape control, shape desgn, shape representaton and shapes preservaton are mportant areas for graphcal representaton of data.see [-]. Monotoncty s a prevalng shape property of curve. There are many physcal stuatons that arse from dfferent scences and art here enttes only have a meanng hen ther values are monotone.examples nclude approxmatons of couple and quas couples n statstcs, approxmaton of potental functons n physcal and chemcal systems and does response curves n bochemstry and pharmacology. The specfcaton of certan devces lke dgtal-to-analog (DAC- used n audo/vdeo devces) and analog-to-dgtal (ADC- used n musc recordngs, dgtal sgnal processng) requres monotoncty, hch seems to be a sort of confuson. In these devces the output drecton s supposed to be the same as that of nput drecton. In terms of monotoncty, as the nput to the devce ncreases (decreases) the output must also ncrease (decrease) accordngly. Monotoncty of monotone data s also nvolved n some other areas lke, the level of blood urc acd n gout patents, data generated from stress and stran of a materal, graphcal dsplay of Neton's la of coolng, medcal dagnoss and economc forecastng. Postvty s a prevalng shape property of curves and surfaces. Postvty preservng problems occur n vsualzng a physcal quantty that cannot be negatve hch may arse f the data s taken from some scentfc,socal or busness envronments.there are some physcal quanttes hch are alays postve: In the modelng of earth surface, measurement of alttude above sea level at dfferent postons on the surface of earth, carbon datng used to measure the age of mummes and fossls, terran modelng, formaton of geologcal crust movement to forecast earth quake and volcanc eruptons, the rate of AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 79

2 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp dssemnaton of drugs n the blood, deprecaton of the prce of computers, probablty dstrbutons and resstance offered by an electrc crcut. Bézer curves and surfaces are the basc tools for modellng n Computer Aded Geometrc Desgnng (CAGD) and Computer Graphcs (CG). The Problem of shape preservaton has been dscussed by a number of authors. In recent years a good amount of ork has been publshed [-] that focuses on shape preservng curves and surfaces.in many nterpolaton problems t s mportant that the soluton preserves some shape propertes such as convexty or monotoncty. Classcal methods usually gnore these knds of condton and thus yeld soluton exhbtng undesrable nflectons. Ths s the reason hy many nvestgatons durng the last years have been drected toards nterpolaton by means of shape- preservng polynomal splne functons. Durng the last fe years, a major research focus has been the use of trgonometrc functons or the blendng of polynomal and trgonometrc functons. Trgonometrc B-splnes ere frst presented n [] and there recurrence relaton for the trgonometrc B-splnes of arbtrary order as establshed n []. An extenson of the Bézer model s studed n [8]. In [9], [], and [] quartc and cubc trgonometrc Bézer curve respectvely th shape parameter s presented and the effect of shape parameter s studed. In ths sequence a ne Quartc ratonal trgonometrc Bézer Curve th a shape parameter s presented. The purpose of ths paper s to present ne ratonal quartc trgonometrc polynomal blendng functons, hch are useful for constructng ratonal quartc trgonometrc B ezer curves and can be appled to construct F contnuous shape preservng nterpolaton splne curves th shape parameters. The changes of a local shape parameter ll only affect to curve segments. In ths paper a ne knd of ratonal quartc trgonometrc Bézer Curve th a shape parameter s presented. The paper s organzed as follos. In secton II, quartc trgonometrc Bézer bass functons are establshed and the propertes of the bass functons are shon. In secton III, ratonal quartc trgonometrc Bézer curves are gven and some propertes are dscussed. By usng shape parameter, shape control of the quartc Trgonometrc Bézer curves s studed. In secton IV, Ratonal Quartc Trgonometrc Parametrc Curve Segments are shon. the representaton of quartc trgonometrc Bézer surface has been shon. In secton V, shape Preservng Interpolaton Splne Curves th local shape parameter are constructed. Numercal examples and applcatons are dscussed n secton VI. Concluson s presented n Secton VII. II. QUARTIC TRIGONOMETRIC BÉZIER BASIS FUNCTIONS Frstly, the defnton of quartc trgonometrc Bézer bass functons s gven as follos. The constructon of the bass functons Defnton. For t [,π/], the follong sx functons are defned as quartc Trgonometrc Bézer bass functons: b = - sn t b = sn t - sn t, b = - sn t - cos t9 + 8 sn t + cos t, b = - sn t- cos t 9 + sn t + 8 cos t, b = cos t - cost, b = - cost, () Fg. shos the curves of the blendng functons. AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

3 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp From the defnton of the blendng functons, We can kno that the blendng functons have the follong propertes analogous to that of the quntc Bernsten bass functons: (a) Non-negatvty: b t, for =,,,,,. (b) Partton of unty: b t =, = (c) Symmetry: b t = b π/ - t, for =,,. - (d) Maxmum: Each III. b t has one maxmum value n [, π/]. RATIONAL QUARTIC TRIGONOMETRIC BÉZIER CURVE.. The constructon of the Ratonal Quartc Trgonometrc Bézer curve: We defne a ratonal quartc trgonometrc Bezer curve as follos: Defnton: Gven ponts P ( =,,,,,) n R orr. For t [, π/], then P b t R t = = b t = () s called a Ratonal Quartc Trgonometrc Bézer curve (RQTB, for short). Where the eght for =,,,,, and the bass functon b t for =,,,,, are defned n Eq. ().Fgure () shos that a Ratonal Quartc Trgonometrc Bezer Curve and Fgure () shos that comparson beteen Quartc Trgonometrc Bezer curve and Ratonal Trgonometrc Bezer Curve Fg. Ratonal Quartc Trgonometrc Bezer Curve Fg. Compare Quartc and RQTB..The Propertes of the Ratonal Quartc TrgonometrcBezer Curve From the defnton of the bass functon, some propertes of the Ratonal Quartc Trgonometrc Bézer curve can be obtaned as follos: Theorem The Ratonal Quartc Trgonometrc Bézer curves () have the follong propertes: Termnal Propertes: R() P, AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

4 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp R( ) P, ' R () (PP ), ' R ( ) (P P ), R () ( 8 )(P P ) (P P ), " R ( ) ( 8 )(P P ) (P P ), " Symmetry: Assume e keep the locaton of control ponts P ( =,,,,) fxed, nvert ther orders, and then obtaned curve concdes th the former one th opposte drectons. In fact, from the symmetry of Ratonal Quartc Trgonometrc Bézer curve, e have R( t, P, P, P,,, ) = (,,,,,,, ) ; [, P P P R t P P P P P P t ], ()Geometrc Invarance: The shape of a Ratonal Quartc Trgonometrc Bézer curve s ndependent of the choce of coordnates,.e. () satsfes the follong to equatons: R( t, P, P, P, P, P, P ) q R( t, P q, P q, P q, P q, P q, P q ) ; t [, ], R( t, P, P, P, P, P, P ) T R( t, P T, P T, P T, P T, P T, P T) ; t [, ], Where q s arbtrary vector n R or R and T s an arbtrary d * d matrx, d = or. ()Convex Hull Property: The entre Quartc Ratonal Trgonometrc Bézer curve segment les nsde ts control polygon spanned by P, P, P, P,P, P. ()Varaton dmnshng property: The curves generated by the normalzed totally postve B-bass have varaton dmnshng property. In ths paper, each of the Ratonal quartc trgonometrc polynomal blendng functons constructed n the space {; sn t; cos t; cos t; sn t; cos t; sn t; cos t} has one maxmum value n [,π/]. The Ratonal quartc trgonometrc B ezer curves le beteen the quntc B ezer curves and the controllng polygon. And the Ratonal quartc trgonometrc B ezer curves are closer to the control polygon than the quntc B ezer curves, hch ndcates that the Ratonal quartc trgonometrc B ezer curves can preserve the feature of the control polygon better than the quntc B ezer curves. (6)Convexty-preservng property: The varaton dmnshng property means the convextypreservngproperty holds. IV. Ratonal Quartc Trgonometrc Parametrc Curve Segments Gven the nterpolaton ponts V ( = ; ; ; ) and the ratonal quartc trgonometrc B ezer control ponts P( = ; ; ; ; ; ), allong for the contnuty and the shape preservng property, the termnal ponts requrements are gven n the follong: R() P V, R ' () (P P ) (V V ), " R () ( 8 )P (8 )P ( )P (V V V ), () " R ( ) ( )P (8 )P ( 8 )P (V V V ), ' R ( ) (P P ) (V V ), R( ) P V,, [, ], here e call, as shape parameters. AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

5 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp The curve segment can be generate usng Eq. () and the blendng functons, as follos: PROPOSITION. : Let, [, ], be the shape parameters, P( = ; ; ; ; ; ), the ratonal quartc trgonometrc B ezer control ponts and V( = ; ; ; ) the correspondng nterpolaton ponts, then for t [, π/] e have b t P Q(t,, ) = = Tb t,, V, = b t = () Where Tb t,, b t b t, b t = Tb t,, b t b t b t b t b t, 6 b t () = Tb t,, b t b t b t b t b t, 6 b t = Tb t,, b t b t, b t = It s easy to proof that Tb t,,. V. Shape Preservng Interpolaton Splne Curves Gven nterpolaton ponts V R d (d = ; ; = ; ; ; n), knot vector U = (u ; u ; ; u n ),and shape parameters α ( = ; ; ; n ), here, u < u < < u n, and α [,+ ).For = ; ; ; n, the -th ratonal quartc trgonometrc parametrc curve segment s j t π/, (6) j Q (t,, ) Tb (t,, )V, Where Tb (t,, ) s gven n Eq. (). The correspondng ratonal quartc trgonometrc parametrc splne curve composed by all of the trgonometrc parametrc curve segments are defned as follos: u u Q u Q (.,, ),u u,u u (7) Where u u u and =,, n. Theorem : The splne curve Q(u) has F contnuty at the nner knots u ; = ; ; ; n. u u Proof: Consder the contnuty at the knot u +. For u u,u, t., e have u AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

6 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp k k k Q u. Q t,,,k,,,, u By smple calculatons e have Q u Q u Q u Q u Q u u u Q u Q u u u ( u ) Q u Q u Q u u ( u ) (8) From here, the theorem follos at the knot u +. We can deal th other knots n the same ay. From Eq. (7) and Eq. (8), t s clear that Q(u) nterpolates the nterpolaton ponts Q ( =; ; ; n ). To generate an open curve Q(u) nterpolatng all of the ponts Q ( =; ; ; ; n), e can add to control nterpolaton ponts Q ; Q n+, to knots u ; u n, and to shape parameters α ; α n. For generatng a closed curve Q(u) nterpolatng all of the ponts P ( = ; ; ; ; n), e can add three nterpolaton ponts Q = Q n; Q n+ = Q ; Q n+ = Q, three knots u ; u n; u n+, and three shape parameters α ;α n;α n+. VI. Numercal Examples and Applcatons Takng nto account the varous applcatons of B-splne functons, t can see that the propertes of B-splne functons mentoned above can be useful n solvng some problem related to approxmaton theory, numercal analyss or computer graphcs, for example representaton of splnes. To compare our computed results and justfy the accuracy and effcency of our presented trgonometrc functons e consder the follong examples. Fgure and Fgure sho open and closed trgonometrc polynomal planar curves generated by usng the shape preservng trgonometrc Interpolaton splne curves. Fgure 6 and Fgure 7 sho that glass model and Floer model Fg. () Open curve Fg. () Closed curve AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

7 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp Fg. (6) Wne glass model Fg. (7) Floer model VII. Concluson As mentoned above Ratonal Quartc Trgonometrc Bézer curve has all the geometrc propertes that classcal quartc Bézer curves have. The shape of the curve can be flexbly controlled by the shape parameter thout alterng the control ponts. Snce there s nearly no dfference n structure beteen a Ratonal Quartc Trgonometrc Bézer curve and a classcal quartc Bézer curve, t s not dffcult to adapt a Ratonal Quartc Trgonometrc Bézer curve to a CAD/CAM system that already uses the classcal quartc Bézer curves.the Ratonal quartc trgonometrc polynomal blendng functons constructed n ths paper have the propertes analogous to that of the quartcbernsten bass functons. And the Ratonal quartc trgonometrc B ezer curves are also analogous to the quartc B ezer curves. Specally, Ratonal quartc trgonometrc B ezer curves are closer to the control polygon than the quartcand quntc B ezer curves. Therefore, theratonal quartc trgonometrc B ezer curves can better preserve the shape of the control polygon. For any shape parameters satsfyng the shape preservng condtons, the obtaned shape preservngratonal quartc trgonometrc nterpolaton splne curves are all F contnuous. Although the shape preservng property s dscussed on planar curves, the numercal example ndcates that our method can be also appled to generate nce feature preservng space curves. References. J. Schoenberg, 96, On Trgonometrc Splne Interpolaton, J. Math. Mech., Vol., pp T. Lyche, R. Wnther, 979, A Stable recurrence relaton for trgonometrc B-splnes, J. Approx. Theory, Vol., pp X. Han,, Quartc Trgonometrc B_ezer Curves and ShapePreservng Interpolaton, Journal of Computatonal Informaton Systems 8: () 9 9. X. Han,, Cubc Trgonometrc Polynomal Curves th a Shape Parameter, Computer Aded Geometrc Desgn, Vol., pp X. Han, 6, Quadratc trgonometrc polynomal curves concernng local control, Appled Numercal Mathe-matcs, Vol. 6, pp X. A. Han, Y. C. Ma, X. L. Huang, 9, The Cubc Trgonometrc Bézer Curve th To Shape Parameters, Appled MathematcalLetters, Vol., pp X-An Han, YChen Ma, XL Huang, 8, A novel generalzaton of Bézer curve and surface, Journal of Computatonal and AppledMathematcs, 7, pp Cheng We Wang, Jul.7, Cubc Trgonometrc Polynomal Splne Curves th Shape Parameter, Journal of Bejng Insttute of Clothng Technology, vol.8, pp Benyue Su and Youdu Huang,, Propertes and Applcatons of T-B Splnes, College Mathematcs, (), pp.87-9 (n Chnese).. Xul Han,, A class of general quartc splne curves th shape parameters, Computer Aded Geometrc Desgn, 8, pp Lanlan Yan, Jongfeng Lang,, A Class of Algebrac-Trgonometrc Blended Splnes, Journal of Computa-tonal and AppledMathematcs,, pp Huayong Lu, Lu L, Damng Zhang,, Study on a Class of T-C Bézer Curve th Shape Parameters, Jour-nal of Informaton andcomputatonal Scence, 8: 7, pp We Xang Xu, Lu Qang Wang, Xu Mn Lu,, Quadratc TC-Bézer Curves th Shape Parameter, Advanced Materals Research,vols. 79-8, pp AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 8

8 Mrdula Dube et al., Amercan Internatonal Journal of Research n Scence, Technology, Engneerng & Mathematcs, 9(), June-August, 7, pp L.L. Yan, X.B. Yang, L.Z. Song, 8, Bézer Curves th To Shape Parameters, Journal of Engneerng Graphcs, vol.9, pp Xaoqn Wu, Xul Han, 7, Cubc Trgonometrc Polynomal Splne Curves th a Shape Parameter, Com-puter Applcatons andsoftare, vol., pp XaoqnWu, Xul Han, Shanmng Luo, 8, Quadratc Trgonometrc Polynomal Bézer Curves th a Shape Parameter, Journal ofengneerng Graphcs, vol.9, pp X-An Han, XL Huang, YChen Ma,, Shape Analyss of Cubc Trgonometrc Bézer Curves th a Shape Parameter, AppledMathematcs and Computaton, 7, pp L.Yan and J. Lang, An extenson of the Bézer model, Appled Mathematcs and Computaton, vol. 8, no. 6, pp Mrdula Dube, Reenu Sharma,, Quartc Trgonometrc Bézer Curve th a shape parameter, Internatonal Journal of Mathematcsand Computer Applcatons Research (IJMCAR), Vol., Issue, Aug, pp Qn Xnqang, Shen Xaol, Hu Gang, Shape modfcaton for quartc C-Bézer curves [J]. Computer Engneerng and Applcatons,,(): Y. Zhu, X. Han and J. Han, Quartc Trgonometrc Bézer Curves and Shape Preservng Interpolaton Curves, Journal of ComputatonalInformaton Systems, 8 (),() 9-9. AIJRSTEM 7- ; 7, AIJRSTEM All Rghts Reserved Page 86