C 1 Positive Bernstein-Bézier Rational Quartic Interpolation Maria Hussain, Malik Zawwar Hussain and Maryam Buttar

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1 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 C Postve Bernsten-Bézer Ratonal Quartc Interpolaton Mara Hussan Malk Zawwar Hussan and Maryam Buttar Astract A postvty preservng C ratonal quartc Bernsten-Bézer nterpolaton scheme s developed for scattered data. The constrants are developed on the Bézer ordnates to preserve the postve shape of scattered data. The weght functons are free to mprove the shape of the surface. The developed scheme has more degrees of freedom as compared to exstng ratonal postvty preservng nterpolaton schemes. It s local and equally applcale to data as well as data wth dervatves. Keywords- C Bernsten-Bézer ratonal quartc functon postvty scattered data nterpolaton weght functons Bézer ordnates. I. INTRODUCTION Scattered data orgnates from a numer of applcatons ncludng earth scences meteorology electronc magng reverse engneerng ndustral desgn 3D photography shp uldng aeronautcs and medcal modellng of human ody parts. The data can e convex monotone or postve partally or over the whole doman. However the prolem under consderaton here s postvty of scattered data. The prolem of postvty preservng nterpolaton of scattered data has een dscussed y many authors. Some of the notceale contrutons are revewed here. Malk Zawwar Hussan s a Professor n Department of Mathematcs Unversty of the Punja Lahore Pakstan (e-mal: malkzawwar.math@pu.edu.pk) Mara Hussan s an Assstant Professor n Department of Mathematcs Lahore College for Women Unversty Lahore-Pakstan(e-mal: marahussan_@yahoo.com) Maryam Buttar s a M.Phl. Student n Department of Mathematcs Unversty of the Punja Lahore Pakstan Brodle Asm and Unsworth [] dscussed the prolem of vsualzaton of data y usng modfed quadratc Shepard method. To assure the postvty of nterpolant suffcent condtons were mposed on quadratc ass functons. To nterpolate the fractonal data they nhted the nterpolant wthn the lmts [0]. The generalzed form of the nterpolant was produced. In [6] Hussan and Hussan developed a postvty preservng nterpolaton scheme whch was ased on Coons patches. A ratonal functon wth shape parameters (Hermte form) was used to nterpolate each oundary of trangle. The sde-vertex nterpolaton was also performed y a ratonal functon (Hermte form). The fnal surface patch was the convex comnaton of these sde-vertex nterpolant. The authors n [6] developed data dependent constrants on shape parameters to preserve the postve shape of data. Hussan and Hussan [7] developed a C Bernsten-Bézer ratonal cuc scheme to preserve the postve shape of scattered data. Luo and Peng [8] presented a range restrcted C ratonal splne nterpolaton scheme for scattered data. The ratonal splne nterpolant was represented as a convex comnaton of three cuc Bézer trangular patches. The range restrcted ostacles were the polynomal surfaces of degree up to three. Mulansky and Schmdt [9] proposed a range restrcted C quadratc splne nterpolaton scheme for scattered data. The doman was trangulated y the Powell-San refnement. The soluton to the prolem came out as a solvale system of lnear nequaltes for the gradents as parameters. The nequaltes were solved y a gloal quadratc optmzaton prolem the thn plate functonal was the ojectve functon and the system of nequaltes were the constrants. Herrmann Mulansky and Schmdt [] proposed a unvarate and varate C quadratc range restrcted nterpolaton scheme for scattered data. In [9] the range ISSN:

2 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 restrctng constrants were pecewse constants as n [] the pecewse quadratc functons were the lower and upper ostacles to the value of nterpolant. In ths paper a C Bernsten-Bézer ratonal quartc nterpolaton scheme s developed to preserve the postve shape of scattered data. The ratonal quartc Bernsten-Bézer functon has 5 Bézer ordnates and 5 weght functons. Due to hgh numer of Bézer ordnates(control ponts) the nterpolated surface s closer to the shape of control polygon as compared to lesser degree nterpolatng functons. The weght functons provde degrees of freedom for surface s shape mprovement f desred. The constrants are developed on the Bézer ordnates to preserve the postve shape of scattered data. If for any trangular patch the Bézer ordnates fal to oey the developed constrants then weghts functons are modfed to preserve postve surface through postve scattered data. The advantages of the proposed ratonal quartc nterpolaton scheme to the exstng ones are as follows: In [] [8] and [9] constrants were developed on the dervatves at knots to assure postvty. The proposed postvty preservng ratonal quartc nterpolaton scheme of ths paper keeps the dervatves unconstraned hence equally applcale to data as well as data wth dervatves. In [7] a postvty preservng ratonal cuc Bernsten-Bézer nterpolaton scheme was proposed. The ratonal cuc functon of [7] had ten Bézer ordnates as the ratonal quartc scheme of ths paper has ffteen Bézer ordnates. It follows that the postve surface developed y ratonal quartc Bernsten-Bézer functon s closer to the control polygon as compare to the ratonal cuc nterpolant [7]. Moreover t has ffteen weght functons ut cuc nterpolant [7] had ten weght functons. Thus ratonal quartc nterpolant has more degrees of freedom as compared to [7]. Unlke [] the proposed ratonal quartc nterpolaton scheme of ths paper s local. The remanng paper s organzed as follows: The Secton II presents the Bernsten-Bézer ratonal quartc functon. The prolem of postvty and C contnuty are dscussed n Secton III and Secton IV respectvely. Secton VI concludes the paper. II. BERNSTEIN-BÉZIER RATIONAL QUARTIC UNCTION The Bernsten-Bézer functons can e used to nterpolate the scattered data only f the data s arranged over the trangular grd (doman s unon of non-ntersectng trangles). The ratonal Bernsten-Bézer functons possess free parameters known as weght functons. The smoothness of nterpolated surface s olged to sutale choce of weght functons. The Bernsten-Bézer ratonal quartc functon s defned as: P ( uvw ) Puvw ( ) () P uvw P uvw j k uvw ( ) P u v w jk + j+ k j k + j+ k jk u v w. jk + j+ k are the weght functons and 00 are the Bézer ordnates at the three vertces of trangle; and 30 are the oundary Bézer ordnates; are the nner Bézer ordnate and u v w are the arycentrc coordnates. The arycentrc coordnates wth respect to the trangle VVV 3 can e computed from the followng convex comnaton: V uv + vv + wv3 u+ v+ w and u > 0 v> 0 w> 0. jk V( x y) s an artrary pont of the trangle wth vertces V ( x y ) V ( x y ) and VVV 3 V x y ISSN:

3 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 III. POSITIVE BERNSTEIN-BÉZIER RATIONAL QUARTIC UNCTION In ths secton the suffcent condtons are derved on the ordnates of Bernsten-Bézer ratonal quartc functon () to preserve the shape of postve scattered data. Puvw> 0f It can e easly oserved that P( u v w ) > and P ( u v w ) > 0 0. We assume that the Bézer ordnates at the vertces are strctly postve.e. 00 > 0 00 > 0 and 00 > 0. The suffcent condtons on the remanng Bézer ordnates are derved to ensure the postvty of the entre patch r (3) r > 0. The weght functons are chosen n the followng way: 00 l 00 m 00 n jk t + j+ k j k () l > 0 m > 0 n > 0 and t > 0. P uvw depends upon the Bézer ordnates jk j k + j+ k. Susttutng the values of Bézer ordnates and weght functons from ()-() nto P ( uvw ) t s rewrtten as: The postvty of + + P uvw ull vmm wnn ( rt u v + uv + u w + uw + v w + vw uv uw vw uvw uvw ) + uvw. (5) g. Dstruton of the Bézer ordnates of the Bernsten-Bézer ratonal quartc functon. Snce u v + uv + u w + uw + v w + vw + 6u v + 6uw + 6vw + uvw+ uvw+ uvw u v w e 3 V e Utlzng the aove expresson n (5) P ( uvw) takes the form P uvw u ll+ rt + v mm+ rt w ( nn rt) rt. + + (6) V e V 3 Let g. Locaton of edges of VVV L 00 M 00 N () L > 0 M > 0 and N > 0. It s assumed that all the oundary Bézer ordnates have the same value.e. rom (5) t s clear that when r 0 P( u v w ) > when rt ncreases P ( uvw) 0 decreases. Assume that l m n r and t are fxed at the mnmum value of P ( uvw ) the followng condtons hold: P P u v 0 (7) ISSN:

4 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 P P u w 0. Susttutng the value of (8) P uvw from (6) n to (7) and (8) the followng rato s otaned u: v: w : :. ll + rt 3 mm + rt 3 nn + rt 3 (9) Snce u+ v+ w the rato nvolved n (9) provdes the followng values of arycentrc coordnates u v and w u v 3 3 S ll+ rt S mm + rt w. S 3 nn + rt S ll + rt mm + rt nn + rt Susttutng these u v and w nto (6) we otan the mnmum value of P ( uvw ). Puvw ( ) rt ll + rt mm + rt nn + rt (0) It s clear from (0) P ( u v w ) > 0 when r 0 and P ( uvw) decreases as r ncreases. Equaton (0) s rewrtten as: P ( u v w) rt R ll mm nn R rt rt rt Replacng y m > 0 r n () we have t P ( u v w) m R 3 () () 3 mll mmm mnn R t t t P uvw we shall estmate mnmum value of r. Solvng m Equaton () for m we have To assure postvty of m ( m ) (3) + +. mll 3 mmm 3 mnn t t t It s oserved that for l > 0 m > 0 n > 0 L > 0 M > 0 N > 0 t > 0 and m > 0 we have ( m ) < 0 and m m > Snce m 0. m > 0 s convex over the whole doman. Ths oservaton reports that ( m ) must have one mnma over the entre nterval. Solvng the Equaton (3) we shall fnd the requred value of m. The Equaton (3) can solved from any numercal technque of estmaton of roots however here t s solved through Newton- Raphson method. The requred lower ound of Bézer ordnates and s r. m The aove dscusson can e summarzed as: Theorem. The Bernsten- Bézer ratonal quartc functon () wth 00 L 00 M 00 N L > 0 M > 0 and N > 0 s postve ( Puvw> ( ) 0 uvw ) f the Bézer ordnates satsfy the relaton r r s otaned y solvng the Equaton (3) r. m ISSN:

5 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 IV. C POSITIVE BERNSTEIN-BÉZIER RATIONAL QUARTIC SURACE The constructon of C postvty preservng surface for the scattered data ( x y z) 3... N z > 0 nvolves the followng steps:. Trangulaton of the doman data ( x y ) 3... N.. Intalzaton of partal dervatves. 3. Creaton of the fnal surface whch comprses of quartc Bézer trangular patches each of whch s guaranteed to reman postve. The Bernsten-Bézer ratonal quartc trangular patch s C at the vertces of the trangle f the oundary Bézer ordnates are: uuuur + VV f uuuur VV f uuuur + VV f uuuur VV f uuuur + VV f uuuur VV f V V V3 V3 V V () The values of oundary Bézer ordnates are calculated y adoptng the approach of [7]. or an artrary value of weght functon the oundary Bézer ordnates defned n () may not oey the lower ounds developed n Theorem. rom () t s clear that the postvty condton proposed n Theorem wll e satsfed y the oundary Bézer ordnates ether y ncreasng the weghts or y decreasng the weghts 00 and. 00 In such a way 00 the derved postvty condtons are satsfed e.g. 03 > r or 00 f f 00 + ( x3 x ) ( V ) ( y3 y ) ( V ) r. + > 03 x y The weghts are nvolved n the postvty preservaton of data and 0 are local to each trangle thus the change n the values of these weghts wll not affect the C contnuty at the vertces. A. Inner Bézer ordnates or each trangle now we wll fnd the nner Bézer ponts. Ths needs to e done n order to ensure C contnuty across patch oundary. In ths scheme each trangular surface patch s a convex comnaton of three quartc Bézer patches. The fnal surface patch s the convex comnaton of the trangular patches P 3. The trangular patches have same vertex and oundary Bézer ordnates ut dfferent nner Bezer ordnates. The constrants are determned on the nner Bézer ordnates to ensure C contnuty along the edges e 3 of trangle. The fnal surface patch P s the convex comnaton of trangular patches P 3. The fnal surface P s expressed as: 3 P CP + CP + C3 P (5) R + R P R + R 3 R + R P P 3 D D D R u v w uv uv uw uw vw vw uv00 + 6uw vw00 R u vw + uv w + uvw R u vw + uv w + uvw R u vw + uv w + uvw 3 ISSN:

6 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 D u + v + w + u v + uv uw30 + uw03 + vw03 + vw03 + 6uv0 + 6uw0 + 6vw0 + uvw + uvw + uvw vw C vw + uw + uv C uw vw + uw + uv C uv vw + uw + uv 3. Susttutng the values of P P and P 3 we have: R+ u vw + uv w + uvw Puvw ( ) D (6) 3 vw + uw + uv vw + uw + uv 3 vw + uw + uv vw + uw + uv 3 vw + uw + uv. vw + uw + uv To determne the values of and we P use the value of normal dervatves n 3 along the edges e 3. Hence the nward normal dervatve of P.e. P along the e s n P P P P () + + n u v w P P P. u v w (7) n s the nward normal to the edge e. Smlarly the nward normal to the remanng edges are n and n3. P P Susttutng the values of and P n u v w (7) and after some smplfcaton we have P Av + Bvw+ Cvw+ Dvw+ Evw+ vw+ Gvw+ Hw n av v w cv w dvw ew 3 3 ( ) A ( ) 00 B ( ( ) (8) C ( ) 00 ( ) D ( ) 00 ( ) 03 + ( ) 03 + ( ) 0 E ( ) 00 + ( ) 03 + ( ) 03 ( ) ( ) 00 + ( ) ) ISSN:

7 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 G ( ) 00 + ( ) 03 H Snce we are nterested n the lnear value of P (oth numerator and denomnator of (8) n should e lnear). Numerator of (8) s made lnear y the followng susttutons: B 6 A+ H C 5A+ 6 H D 0A+ 5 H E 5A+ 0 H 6A + 5 H G A + 6 H. Smlarly for the denomnator a c 6 0 d 03 e. 00 The expresson av + v 3 w+ cv w + dvw 3 + ew s lnear f c ae a + e c a e.e The aove computaton leads to the followng system of equatons: rom the aove dscusson t s clear that only three parameters ( ) are free whle the values of remanng are dependent on these parameters. Let k k s any postve real numer. Smlarly the normal dervatves along the other P P edges.e. and can e computed. n n3 Lnearty of normal dervatves along the edges provdes the followng set of restrctons on the Bézer ordnates and weght functons. k k 6 k A (9) ' 00 0 k + 8k k00 ' 30 k0 A (0) 8k + 7k + 7k0 ' 8k03 + k 0 A3 () 8k 8k03 + 7k ' 7k0 k 0 A () k + 8k 30k 0 8k03 ' k00 A5 (3) ' k k00 6 k 0 A6 () ' k k00 6 k 0 B (5) 8k + k k00 ' 8k30 30 k 0 B (6) k + 8k 7k0 ' 7k0 k 0 B3 (7) 8k + 7k 8k03 ' + k0 B (8) 8k k00 8k03 ' 30 k0 B5 (9) ' k 00 B6 (30) 3 ' k + k00 6 k 0 C (3) 3 3 k + 8k 8k30 ' k00 30 k 0 C (3) 3 3 7k + 8k 8k30 ' 7k0 + 3 k 0 C3 (33) 8k k 8k + k 30 0 ISSN:

8 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume k C (3) k 8k 8k k 30 k C (35) k k 6 k C (36) ' ' ' ' ' ' A B and C...6 are known(expressons of known Bézer ordnates and ). 00 The set of equatons (9)-(36) has lesser unknowns than numer of equatons. It provdes a flexle choce of unknowns. The optmal choce of oundary Bézer ordnates and weghts are those whch estalsh frst dervatve contnuty at vertces and edges and nherts postve shape of data as postve surface. V. NUMERICAL EXAMPLES In ths secton the results developed n Secton II-Secton IV are mplemented on two postve scattered sets taken n Tale and Tale. rstly the data sets are arranged y Delaunay trangulaton method. g. 3 and g. 6 are the Delaunay trangulaton of the data sets of Tale and Tale respectvely. The lnear nterpolaton of these postve data sets are shown n g. and g. 7 whch clearly ndcate that shape of surface s postve over the whole doman. The nterpolaton of these data sets y the scattered data nterpolaton scheme developed n Sectons II-IV shown n g. 5(Tale ) and g. 8(Tale ). It s clear from g. 5 and g. 8 that postve shape of the surface s preserved. Tale. A postve scattered data set. x y x y x y x y x 0 y x y x y x y Tale. A postve scattered data set. x y x y x y x y x y x y x y x y x y x y x y x y ISSN:

9 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 g. 3 Trangulaton of the doman of postve data set n Tale. g. 6 Trangulaton of doman of postve data set n Tale. g. Lnear nterpolaton of postve data set n Tale. g. 7 Lnear nterpolaton of postve data set n Tale. g. 5 Postve ratonal quartc surface. g. 8 Postve ratonal quartc surface. VI. CONCLUSION In ths study an nterpolaton scheme s developed to preserve the shape of postve scattered data. The Bernsten-Bézer ratonal quartc functon s used as nterpolaton tool. The Bernsten Bézer ratonal quartc functon s the comnaton of Bézer ordnates and weght functons. These weght functons ehave lke ISSN:

10 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 shape refnement parameters (free parameters). The used nterpolaton technque requres pretrangulaton of data. The data s trangulated such that each data pont s lyng at the vertex of trangle. Data ponts provde the values of vertex Bézer ordnates. The values of oundary Bézer are computed y C contnuty at the vertces of trangles. The C contnuty along the edges of trangles provdes the value of remanng Bézer ordnates and weght functons. The lower ounds of nner and oundary Bézer ordnates are determned to assure postvty of trangular patch. If these Bézer ordnates do not oey the developed constrants of postvty then these are modfed y the values of weght functons to assure postvty. The developed nterpolaton s local C applcale to oth data and data wth dervatves and provdes opportunty of shape refnement y sutale choce of weghts. The ratonal structure of nterpolant ncorporates data havng sngulartes. REERENCES [] M. Bastan-Walther and J. W. Schmdt Range restrcted nterpolaton usng Gregory s ratonal cuc splnes Journal of Computatonal and Appled Mathematcs Vol pp.-37. [] K. W. Brodle M. R. Asm and K. Unworth Constraned vsualzaton usng Shepard nterpolaton famly Computers and Graphcs orum Vol. No. 005 pp [3] T. N. T. Goodman H. B. Sad and L. H. T.Chang Local dervatve estmaton for scattered data nterpolaton Appled Mathematcs and Computaton Vol pp [] M. Herrmann B. Mulansky and J. W. Schmdt Scattered data nterpolaton suject to pecewse quadratc range restrctons Journal of Computatonal and Appled Mathematcs Vol pp [5] Øyvnd Hjelle and M. Dæhlen Trangulatons and Applcatons Sprnger (006). [6] M. Z. Hussan and M. Hussan C postve scattered data nterpolaton Computers and Mathematcs wth Applcatons Vol pp [7] M. Z. Hussan and M. Hussan C postvty preservng scattered data nterpolaton usng ratonal Bernsten-Bézer trangular patch Journal of Appled Mathematcs and Computng Vol. 35 No. - 0 pp [8] Z. Luo and X. Peng A C ratonal splne for range restrcted nterpolaton of scattered data Journal of Computatonal and Appled Mathematcs Vol pp [9] B. Mulansky and J. W. Schmdt Powell- San splnes n range restrcted nterpolaton of scattered data Computng Vol. 53 No. 99 pp M. Z. Hussan s a Professor n Punja Unversty Lahore Pakstan. He receved hs Ph.D. from Punja Unversty Pakstan n 00. Hs research nterests nclude Computatonal Mathematcs Computer Graphcs CAGD CAD/CAM Image Processng and Soft Computng. He has advsed/supervsed more than 36 students for ther M.Phl. and PhD theses. He has pulshed more than 65 pulcatons n the form of journal and conference papers. M. Hussan s an Assstant Professor n Lahore College for Women Unversty Lahore Pakstan. She receved her Ph.D. from Punja Unversty Pakstan n 009. Her research nterests nclude CAGD CAD and Vsualzaton. She has supervsed more than 3 students for ther M.Phl theses. She has pulshed more than 30 pulcatons n the form of journal and conference papers. M. Buttar s a research student n Unversty of the Punja Lahore Pakstan. ISSN:

11 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 Tale 3. Numercal results of g x V V y x V V y x V3 V3 y ISSN:

12 INTERNATIONAL JOURNAL O MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8 0 Tale. Numercal results of g x V V y x V V y x V3 V3 y ISSN:

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