Convexity Preserving C 2 Rational Quadratic Trigonometric Spline

Transcription

1 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 ISSN Covexy Preservg C Raoal Quarac Trgoomerc Sple Mrula Dube, Pree Twar Deparme of Maemacs a Compuer Scece, R. D. Uversy, Jabalpur, Ia. Deparme of Maemacs, G.G.I.T.S., Jabalpur, Ia. Absrac- A C raoal quarac rgoomerc sple erpolao as bee sue usg wo ks of raoal quarac rgoomerc sples. I s sow a uer some aural coos e soluo of e problem exss a s uque. Te ecessary a suffce coo a cosra e erpola curves o be covex e erpolag erval or suberval are erve. approxmao properes as bee scusse a cofrms e expece approxmao orer s. Iex Terms- Approxmao, Cosrae erpolao, Couy, Covexy, Raoal quarac rgoomerc sple, Sape parameer. D I. INTRODUCTION urg e rece years raoal paramerc sple ave gae wesprea accepace for use compuer ae geomerc esgs.tey ave bee sue by several auors see ], ], 8], ], w specal empass o e sape preservg properes. I ] Dua ave presee e cosruco a sape preservg aalyss of a ew wege raoal cubc erpolao a s approxmao. Trgoomerc sples serves as a alerave for polyomal sples for solvg may problems of erpolao a ave bee sue from applcao po of vew see ], 3], 5], 6], 7], 9]. Trgoomerc sples beave a beer way for pa approxmao or scaere aa erpolao. B-sples rouce by Scoeberg 9], ave become mmesely popular for applcaos curve a surface geerao problems.keepg a vew of e above eas a e applcaos of rgoomerc sples, we ave exee e eas of Dua ] a cosruce covexy preservg raoal rgoomerc sple. We ave rouce a wege raoal quarac rgoomerc sple erpolao a e C couy of raoal rgoomerc sple.te covexy corol a approxmao properes of C raoal quarac rgoomerc sple ave bee scrbe. II. A C WEIGHTED RATIONAL QUADRATIC TRIGONOMETRIC INTERPOLATION A wege raoal cubc sple erpolao base o fuco values a ervave was gve ]. Gve a aa se, f,,,,...,,,were f a are e fuco values a e ervave values efe a kos, respecvely, a < <... < < are e kos. Le,,, ] a, are s f s s U cos cos V P cos s cos f were U f V f Ts raoal quarac rgoomerc sple P sasfes

2 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 ISSN P f, P,,,...,,. for e gve aa se, f,,,...,, le s f s s U, cos cos V P cos s, cos f were U, f V f, wc f f. Obvously e sple P sasfes P f, P ',,,...,. I s calle raoal quarac rgoomerc sple base o fuco values. Te wege raoal quarac rgoomerc sple wll be cosruce by usg e wo ks of raoal rgoomerc quarac sple erpola escrbe above.le were P P P, ],,,...,. 3 a s f s s U cos cos V cos f P cos s U V f, f, R. w e weg coeffce Ts raoal quarac rgoomerc sple P sasfes P f, P,,,...,.

3 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 3 ISSN III. A C WEIGHTED RATIONAL QUADRATIC TRIGONOMETRIC SPLINE INTERPOLATION We ow follow e famlar proceure of allowg e ervave parameers.,...,, o be egrees of freeom wc are cosrae by e mposo of e C couy coos.,...,, ' ' P P e coos leas o e followg couous sysem of lear equaos: } { { } 5,...,-. Terefore, f e successve parameers, a, sasfy 5 a,,,..., amely, for e posve parameers, a e selece, f ]} ] { ] } { e,. P C IV. CONVEXITY CONTROL OF RATIONAL QUADRATIC TRIGONOMETRIC SPLINE Posvy, mooocy a covexy are basc a fuameal sapes, wc ormally arse everyay scefc peomea. To ge coo for e erpolao o keep covex e erpolag erval,coser e coo for e seco orer ervave o rema posve or egave e erpolag erval, s ask ca be carre ou smply by selecg suable values of e parameer o sasfy e lear equaly.i s seco we assume a e kos are equally space. For smplcy of preseao le us assume a srcly covex se of aa so a. < <... < I a smlar faso, oe ca eal w a cocave aa so a. > >... >

4 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 ISSN For a covex erpola P, s e ecessary a e ervave parameers soul be suc a < <... < < <... < < a for cocave aa. > >... > > >... > > Now P s covex f a oly f ' P For, ] ', e seco-orer ervave P ca be compue a as e form ' 3 P Z. cos s 6 were Z Q R T a Q cos s { s U f s s V f cos cos U f V f cos V f s U f V f cos U f V U f} R cos s s cos {cos U s cos V f U f s f V } f T s cos { s f s s U cos cos V cos f }

5 P Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 5 ISSN Te suffce a ecessary coo for e erpolag fuco P efe by o be covex o, ] parameer sasfy s e posve f f 7 Here s observe a f e aa are posve/covex e e erpola wll be posve/covex. Tus we ave prove e followg eorem Teorem.Gve, f,,,,...,,,e ecessary a suffce coo for e erpolao efe by 3 o be covex o, ] s a e gve aa a e posve parameer sasfy f f 8 V. NUMERICAL EXAMPLE Example:Le f cos / 6,.5,.5] w erpolag kos a.5,.5, 3., ,.5,.75. le.99, a le f s e fuco beg erpolae.deoe e correspog C -couous erpolag fuco efe by.5,.5] by p sce e erpolag aa a parameer sasfy e coo of eorem. as sow fgure Fgure : grap of P VI. APPROXIMATION PROPERTIES OF THE WEIGHTED RATIONAL QUADRATIC TRIGONOMETRIC INTERPOLATION To esmae error of e wege raoal quarac rgoomerc erpolag fuco efe by, sce e erpolao s local,wou loss of geeraly, we coser e error e suberval, ]. We f C, ] a P s e raoal quarac rgoomerc sple erpolag fuco of f, ] f space,amely, for all,,...,, usg e Peao-Kerel Teorem Sculz 3 ] gves e followg. I s easy o see a s ype of erpolao s exac for, e polyomal beg erpolae, wc e egree s o more a. Coser e case we e kos are equally R f ] f P f R ],, ] 9

6 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 6 ISSN were were R p < < q < < ] r < < p { s s cos cos cos ] cos cos }/ cos s, < < ; q { s s cos cos cos ] cos cos }/ cos s, < ; < Te cos cos r, < < ; cos s R f ] f P f { p q r } 3 for r for all, ], us r cos cos cos s

7 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 7 ISSN for q, sce q { s s cos cos cos cos cos }/ cos s ] 5 a cos cos cos s q 6 I s easy o see a e roo of q s cos cos s s cos cos cos ] Tus q q q Q ] Q { s s cos cos cos ] cos cos }/ cos s z z 7 cos cos s s cos cos cos z 8 smlarly p q

8 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 8 ISSN p { s s cos cos cos cos cos }/ cos s ] 9 a e roo of p, ] s were M N M cos s cos cos ] N s s cos cos cos cos s ] so a p p p { z z were z 3 cos s cos cos } cos s s s cos cos cos ] cos cos Q z M N ] z M N ] M z 3 ] N

9 Ieraoal Joural of Scefc a Researc Publcaos, Volume 3, Issue 3, Marc 3 9 ISSN R f ] f P f w,, were were w,, s a cosa epeg upo,,. w cos cos,, Q Q ] cos s ACKNOWLEDGMENT We are akful o Prof. Xul Ha Dep. of Apple Maemacs a Apple Sofware, Ceral Sou Uversy, Cagsa, PR Ca for s valuable suggesos. REFERENCES ] Q. Dua, L. Wag a E. H. Twzell, A ew wege raoal cubc erpolao a s approxmao, Appl. Ma. Compu., 5, 68, pp ] M. Dube, R. Sarma, Quarac NUAT B-sple curves w mulple sape parameers, Ieraoal Joural of Mace Iellgece,,3, pp.8. 3] M. Dube, S. Sayal, Sape preservg G raoal cubc rgoomerc sple, Joural of e Ia Ma. Soc.,, 78, pp ] J. A. Gregory, Sape preservg sple erpolao, Compu. Ae Des., 986, 8, pp ] X. A Ha., Ma. Y. Ce, a X. L Huag, Te cubc rgoomerc Bézer curve w wo sape parameers, Apple Maemacs Leers, 9,, pp ] Xul. Ha, Quarac rgoomerc polyomal curves w a sape parameer, Compuer Ae Geomerc Desg,, 97, pp ] S. S. Raa, M. Dube, S. Sayal A GC cubc rgoomerc B-sple, Ivesgao Maemacal Sceces,,, pp ] M. Sarfraz, Covexy preservg pecewse raoal erpolao for plaer curves, Bulle of Korea Maemacal Socey, 99,9, pp.93. 9] I. J. Scoeberg, O rgoomerc sple erpolao, J.Ma. Mec., 96, 3, pp ] M. H. Sculz, Sple Aalyss,Prece-Hall:Eglewoo Clffs, New Jersey, 973. AUTHORS Frs Auor Mrula Dube, P. D., Professor, Deparme of Maemacs a Compuer Scece, R. D. Uversy, Jabalpur, Ia Seco Auor Correspoece Auor Pree Twar, M. Sc., As. Professor, Deparme of Maemacs, G.G.I.T.S., Jabalpur, Ia. Emal aress preewar976@gmal.com Alerae emal aress jaev.war@gmal.com Coac umber