Computer Aded Geometrc Desg 9 79 78 www.elsever.com/locate/cagd Applcato of Legedre Berste bass trasformatos to degree elevato ad degree reducto Byug-Gook Lee a Yubeom Park b Jaechl Yoo c a Dvso of Iteret Egeerg Dogseo Uversty Busa 7-7 South Korea b Departmet of Mathematcs Educato Seowo Uversty Cheogju -7 South Korea c Departmet of Mathematcs Dogeu Uversty Busa -7 South Korea Receved 9 July ; receved revsed form 9 July Abstract We study the relatoshp of trasformatos betwee Legedre ad Berste bass. Usg the relatoshp we preset a smple ad effcet method for optmal multple degree reductos of Bézer curves wth respect to the L -orm. Elsever Scece B.V. All rghts reserved. Keywords: Legedre; Berste; Bass trasformatos; Bézer curves; Degree elevato; Degree reducto. Itroducto We ca express a polyomal curve wth a approprate bass for ts use. The use of orthogoal bass such as Chebyshev ad Legedre polyomal permts optmal degree reducto to exchage covert or reduce data or compare geometrc ettes whch s a mportat task CAGD L ad Zhag 998; Mazure 999. For example we have see the use of Chebyshev ad Legedre polyomal degree reducto schemes Watks ad Worsey 988; Eck 99 99. O the other had the Berste form of a polyomal havg the recursve formula ad the property of partto of uty offers valuable sght to ts geometrcal behavor ad has wo wdespread acceptace as the bass for Bézer curves ad surfaces CAGD Far 99. But Berste polyomals are ot orthogoal. So the bass trasformato s mportat ad has bee studed may ways. Farouk foud the explct form of the bass trasformato betwee Legedre ad Berste bass. * Correspodg author. E-mal address: lbg@dogseo.ac.kr B.-G. Lee. 7-89//$ see frot matter Elsever Scece B.V. All rghts reserved. PII: S7-89-
7 B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 I ths paper we fd the relatoshps betwee the Gram matrx Q ad the Legedre Berste bass trasformato matrx M them ad the Berste Legedre bass trasformato matrx M adthe orthogoal matrx U ad M. We also obta the relatoshps betwee the bass trasformato matrces M M ad the degree elevato matrx T the bass trasformato matrces ad the degree reducto matrx. Ths paper s orgazed as follows. We expla the degree Legedre ad Berste bass ad ther trasformatos Secto. We dscuss the relatoshp amog trasformatos M M egevalues of Gram matrx ad a orthogoal matrx U Secto. We preset the explct method to degree elevato ad degree reducto of Bézer curves Sectos ad.. Legedre ad Berste bass The Bézer represetato uses Berste polyomals as bass fuctos for the lear space of polyomals. I terms of the Berste polyomals of degree B t t t... a parametrc polyomal curve P t of degree > the plae ca be expressed as P t c B t t where the {c } are the set of + cotrol pots. The product of Berste polyomals s m B tbm j t j B +m ad the tegrato s +m +j Bk t dt +. +j t The Legedre polyomals costtute a orthoormal bass that s well suted to least-squares approxmato. To emphasze symmetry propertes they are tradtoally defed o the terval [ +] but for our purpose t s preferable to map ths to [ ]. The Legedre polyomals L t o t [ ] ca be geerated by the explct form L t + / t t t where j!. Ths gves the frst few staces!j! j! L t L t t
B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 7 L t t t + L t 7 t t + t. The orthoormalty of these polyomals s expressed by the relato { fj k L j tl k t dt f j k. Cosder a polyomal P t of degree expressed the degree Berste ad Legedre bass o t [ ]: P t c j Bj t l k L k t. j k We are terested the lear trasformato c j M j kl k j... k that maps the Legedre coeffcets l l...l to the Berste coeffcets c c...c ad ts verse. Wrtg c [c c...c ] t ad l [l l...l ] t we may express ths vector-matrx form as c M l. The we have the followg theorem see Farouk. Theorem. The Legedre polyomal L k t ca be expressed the Berste bass B t B t... B t of degree as L k t k k k + k+ B k t j k + j mjk maxj+k k+ k k k j B j t. The elemets of the matrx M that trasforms the Legedre coeffcets of degree polyomals to the Berste coeffcets accordg to equato are gve for jk by mjk k + M j k. j maxj+k k+ k k k j For Berste to Legedre trasformato matrx M see Farouk. Theorem. The elemets of the verse M j j + M j k + j + +j are gve for jk by j k + k + j j+ k k.
7 B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 Example. [ ] [ M M M 7 7 M 7 7 M ] M 7 7 7 7.. L -orm of the polyomal P We compute the L -orm of a Bézer curve of degree. From these equatos ad we obta the followg computato for the L -orm of the polyomal P wth Berste bass: P c B t dt j c c j B tb j t dt j c c j B j +j t dt c c j +. j +j j +j Let the elemets of the Gram matrx Q of the Berste bass be the Q j j + j... +j The the L -orm of the polyomal P s P ct Q c. Here are some examples of Q. 7 Example. [ Q ] Q Q 7 9 9 7.
B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 7 From the defto of the Gram matrx Q ad the mathematcal ducto all the upper left submatrces of the Gram matrx Q have postve determats. So Q s a real symmetrc postve defte matrx see Lee ad Park 997. Thus t ca be dagoalzed by a orthogoal matrx U.e. U U t whose colum vectors are orthoormal egevectors of Q thats Q U D U t where D s the dagoal matrx wth postve egevalues of the matrx Q. The followg theorem s the drect result from Proposto Lyche ad Scherer. Theorem. From the Gram matrx Q gve by we have Q M M D where λ k + + k k...are egevalues of the Gram matrx Q. From the orthoormalty of Legedre bass we obta the followg computato for the L -orm of the polyomal P wth Legedre bass: P l L t dt l l j L tl j t dt l t l. 8 j From Theorem we get the followg theorem that descrbes the relatoshp amog M M ad D. Theorem. For the Berste to Legedre trasformato matrx M M D M t. Proof. From 7 ad 8 we have we have c t Q c l t l. By the defto of M c t Q c c t M we ca also express the L -orm of the polyomal P as tm c. By Theorem ad 9 we obta Q M D M M tm. Multplyg both sdes by M ad cosderg the traspose of both sdes we complete the proof. 9 The followg theorem eables us to compute U wth the explct forms of M ad D. Theorem. For the orthogoal matrx U of the Gram matrx Q we have U M D.
7 B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 Proof. From Theorem we have Q M D M M D D M t M t. D D M D Ad we ca easly check the orthogoalty of M D M D D M D M t t. M D Ths completes the proof. HerearesomeexamplesofU ad D. Example. [ ] [ U D U U ] D D.. Degree elevato For rasg the degree of Bézer curve by oe wthout chagg the shape of the curve. We ca show that ew vertces c are obtaed from the old polygo by pecewse lear terpolato at the parameter values / + see Far 99. c + c + c...+. + We ca rewrte the formula as a lear system T c c where the + + matrx T s T + +........................... +
B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 7 ad the + vector c ad the + vector c are c c c...c t c c t. c...c We may repeat ths process ad the obta a sequece of cotrol pots. After r degree elevatos we have a lear system T r c c r where the + r + + matrx T r T +r T +r...t + T has elemets r j j T r j...+ r ad j... +r By the orthogoalty of Legedre bass the degree elevato of a polyomal wth Legedre bass s gve by l l l...l t l l l...l t. After r degree elevatos we have a lear system I r l l r where the + r + + matrx I r has elemets { I f j r j f j. After trasformg the Berste coeffcets to the Legedre coeffcets by M the degree elevato by I r ad fdg the Berste coeffcets by M +r we obta the followg theorem. Theorem. The degree elevato matrx T r ca be expressed M I r ad M +r as T r M +r I r M.. Degree reducto Whe we fd the best approxmato the sese of L -orm geeral the degree reducto of Bézer curves address the followg problem. Problem L degree reducto. Let {c } be a gve set of cotrol pots whch defe the Bézer curve c t c B t of degree. The fd aother pot set {b } m defg the approxmatve Bézer curve m b m t b B m t of lower degree m<so that a L -dstace fucto d b m c betwee b m ad c s mmzed.
7 B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 The L -dstace of the two Bézer curves b m ad c s defed as followg: d b m c b m t c t dt m b B m t c B t dt. Usg the matrx T mr we ca elevate the degree of b m from m to wherer m b r T mr b. The the curve b m of degree m s rewrtte as a curve of degree b m t b r t ad the dstace s b r B t d b m c d b r c b r B t c B t dt b r c B t dt. Thus we obta the followg theorem for the L -dstace betwee the Bézer curve b m of degree m ad the Bézer curve c of degree. Theorem 7. The L -dstace betwee the two Bézer curves b m ad c s d b m c d b r c A t Q A where A c T mr b b b b...b m t ad c c c...c t. For developg the method rewrte d bm c. d b m c A t Q A [c T mr b] t Q [c T mr b] c t Q c b t Tmr t Q c + b t Tmr t Q T mr b. Oe method of obtag the vector b s so-called the method of least squares Lee ad Park 997; Lutterkort et al. 999. Ths method cossts of mmzg A t Q A wth respect to b. We choose the vector ˆb as that the value of b mmzes A t Q A. Equatg A t Q A/ b to zero ad wrtg the resultg equatos terms of ˆb we fd that these equatos are T t mr Q T mr ˆb T t mr Q c. They are kow as the ormal equatos. Theorem 8. The + + matrx T t Q T has the followg property: T t Q T Q.
B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 77 Proof. T t Q T j + + l + j +j l l j + l k l+k j k + + l l k k k j l+k k j j.... From Theorem 8 we have T t mr Q T mr Q m. Hece the real symmetrc postve defte matrx T t mr Q T mr s vertble. Provded T t mr Q T mr exsts we have the uque soluto for ˆb ˆb Tmr t Q T t T mr mr Q c. The approxmate curve gve by s the best approxmato wth respect to the L -orm. By the orthogoalty of Legedre bass the degree reducto of a polyomal wth Legedre bass s gve by l l l...l t l l l...l t. After r degree reductos we have a lear system I r l l r where the r + + matrx I r s............ I r....................... After trasformg the Berste coeffcets to the Legedre coeffcets by M the degree reducto by I r ad fdg the Berste coeffcets by M m we obta the followg theorem. Theorem 9. The degree reducto matrx ca be expressed M T t mr Q T t T mr mr Q M m I r M. I r ad M m as For the degree reducto wth Berste bass we ca use the explct matrx forms of M m M ad I r to compute M m I r M gve by the formula ad respectvely. Therefore our method usg the relatoshp of trasformatos betwee Legedre ad Berste bass s a smple ad effcet method for optmal multple degree reductos wth respect to the L -orm. However ths best approxmato does ot geeral terpolate the gve curve at ts edpots. Thus we have to cosder the smoothess of our method for the practcal use.
78 B.-G. Lee et al. / Computer Aded Geometrc Desg 9 79 78 Here s the example gve by the explct matrx form ad we have the same results wth Lutterkort et al. 999. Example Parametrc case. c [ ] t [ ] l M c t c + + [ M I M 9 7 7 7 9 [ ] l I M c t [ c M I M c 88 ] t c +. ] Ackowledgemets Ths work was supported by Dogseo Uversty Dogseo Froter Project Research Fud of ad Korea Research Foudato uder grat KRF-99--DP7. Refereces Eck M. 99. Degree reducto of Bézer curves. Computer Aded Geometrc Desg 7. Eck M. 99. Least squares degree reducto of Bézer curves. Computer-Aded Desg 7 8 8. Far G. 99. Curves ad Surfaces for Computer Aded Geometrc Desg rd Edto. Academc Press Bosto. Farouk R.T.. Legedre Berste bass trasformatos. J. Comput. Appl. Math. 9. Lee B.G. Park Y. 997. Dstace for Bézer curves ad degree reducto. Bull. Australa Math. Soc. 7. L Y.M. Zhag X.Y. 998. Bass coverso amog Bézer Tchebyshev ad Legedre. Computer Aded Geometrc Desg 7. Lutterkort D. Peters J. Ref U. 999. Polyomal degree reducto the L -orm equals best Eucldea approxmato of Bézer coeffcets. Computer Aded Geometrc Desg 7. Lyche T. Scherer K.. O the p-orm codto umber of the multvarate tragular Berste bass. J. Comput. Appl. Math. 9 9 7. Mazure M.-L. 999. Chebyshev Berste bases. Computer Aded Geometrc Desg 9 9. Watks M. Worsey A. 988. Degree reducto for Bézer curves. Computer-Aded Desg 98.