Interval extension of Bézier curve

Transcription

1 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L Iterval exteso of Bézer curve JUNCHENG LI Departet of Matheatcs Hua Uversty of Huates Scece ad Techology Dxg Road Loud cty Hua rovce 47 R CHINA E-al: ljucheg8@6co Abstract: - By extedg defto terval of the classcal Berste bass fuctos to be dyac a class of Berste bass fuctos wth a shape paraeter s costructed ths work The ew bass fuctos are sple exteso of the classcal Berste bass fuctos The the correspodg Bézer-lke curve s geerated o base of the troduced bass fuctos The ew curve ot oly has ost propertes of the classcal Bézer curve but also ca be adjusted by alterg value of the shape paraeter whe the cotrol pots are fxed Because the proposed curve s a polyoal odel of the sae degree ad havg ost propertes of the classcal Bézer curve t has ore advatages tha soe exstg slar odels Key- words: Berste bass fuctos; Bézer curve; the sae degree; shape adjustet; shape paraeter Itroducto As a portat geoetrc odelg tool Bézer curve has bee wdely used Coputer Aded Geoetrc Desg (CAGD) ad Coputer Aded Desg (CAD) However whe the cotrol pots are gve shape of the classcal cubc Bézer curve caot be chaged Wth the developet of geoetrc desg dustry shapes of curves ofte eed to be chaged freely For relevg the default of the classcal Bézer curve the Bézer-lke curves wth shape paraeters have bee pad ore ad ore atteto by ay researchers Because the Bézer curve ca be aturally defed after the bass fuctos are detered Therefore costructg the bass fuctos wth shape paraeters becoes the ost effectve way for establshg Bézer-lke curves wth shape paraeters At preset order to troduce shape paraeters to the bass fuctos of Bézer curve the cooly used ethod has two kds Oe s to costruct o-polyoal bass fuctos wth shape paraeters based o trgooetrc or hyperbolc fuctos such as [-6] Aother s to costruct the hgh-degree polyoal bass fuctos wth shape paraeters by creasg the degree of the classcal Berste bass fuctos such as [7-] Although the Bézer-lke curves geerated by those ethods ca effectvely realze shape adjustet by alterg values of the shape paraeters the structure coplexty s thereupo creased The polyoal Bézer-lke curve wth ultple paraeters of the sae degree [] was a practcal ethod but the curve dd ot have the strct syetry that the classcal Bézer curve has Although the polyoal Bézer-lke curve of degree wth - shape paraeters [] satsfed the sae propertes wth the classcal Bézer curve value rage of the shape paraeters of dfferet order curve are dverse fro each other would cause users wth cofuso Is there a spler ethod for costructg bass fuctos descrbg Bézer-lke curve wth shape paraeters that has ost propertes of the classcal Bézer curve? Ag ths proble the a purpose of ths work s to preset a sple ethod for costructg Berste bass fuctos wth a shape paraeter of the sae degree A class of cubc Berste bass fuctos wth a shape paraeter α aed cubc α-berste bass fuctos s costructed through extedg defto terval of the classcal cubc Berste bass fuctos fro E-ISSN: Volue 4 8

2 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L [] to [ α ] ( < α ) O base of the cubc α-berste bass fuctos the α-berste bass fuctos of degree ( 4) are geerated by recurso property of the classcal Berste bass fuctos The the correspodg Bézer-lke curve aed α-bézer curve s aturally defed o base of the α-berste bass fuctos The proposed α-bézer curve has ost propertes of the classcal Bézer curve ad ts shape ca be adjusted by alterg value of the shape paraeter whe the cotrol pots are fxed The rest of ths paper s orgazed as follows I Secto the α-berste bass fuctos are costructed ad soe propertes of the bass fuctos are gve I Secto the correspodg α-bézer curve s defed Soe propertes of the curve effects of the shape paraeter o the curve ad cotuty of the curve are dscussed A short cocluso s gve Secto 4 The α-berste bass fuctos Costructo of the bass fuctos Geerally the classcal Berste bass fuctos ca be expressed as follows [4]! B ( t) = ( t) t ( )!! where t = The classcal Berste bass fuctos have the followg propertes (a) Noegatvty: B () t ( = ) (b) Noralzato: B () t = (c) Syetry: B ( t) = B ( t) ( = ) (d) ropertes at the edpots: = = B () = B () = = = B () = = B () = = Besdes the classcal Berste bass fuctos have the recurso property as follows B () t = ( t) B () t tb () t where t = ad settg B () t = B () t I order to costruct a class of Berste bass fuctos wth a shape paraeter α a sple deal s to exted defto terval of the classcal Berste bass fuctos fro [] to [ α ] ( < α ) Ispred by the recurso property of the classcal Berste bass fuctos the bass fuctos of degree ( 4) wth a shape paraeter α ca be geerated o base of the cubc bass fuctos Therefore the cubc Berste bass fuctos wth a shape paraeter α are frstly costructed as below Suppose the cubc bass fuctos are expressed as follows f () t f () t f () t f () t = t t t M () where t α < α ad M s a udetered 4 4 atrx Dervato calculus to Eq () the f () t f () t f () t f () t = t t M () Because the cubc bass fuctos are hoped to satsfy the sae propertes wth the classcal cubc Berste bass fuctos at the ed pot therefore let t = ad t = α Eq () ad Eq () respectvely the [ ] = [ ] M [ ] = α α α M () [ ] = [ ] M [ ] = α M Fro Eq () the α α α = M (4) α Solvg Eq (4) the M = 6α 6 (5) α α α α α α α α Takg Eq (5) to Eq () the cubc bass fuctos ca be expressed as follows E-ISSN: Volue 4 8

3 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L 6α f () t = t t t α α 6 f() t = t t t α α f() t = t t α α f() t = t t α α (6) where t α < α The cubc bass fuctos expressed as Eq (6) ca be reparaetrzed to the bass fuctos by b ( u) = f ( αu) ( = ) the b ( u ) ( = ) s defed o a fxed terval u [] whch ca be defed as follows Defto The followg four fuctos of u are called the cubc Berste bass fuctos wth a shape paraeter α (cubc α-berste bass fuctos for short) b( u) = u (α ) u ( α) u b( u) = u 6αu u (7) b( u) = u u b( u) = ( α) u ( ) u where u < α Eq (7) ca be rewrtte as follows b( u) = ( u) ( u) ( α ) u b( u) = u( u) (8) b( u) = u ( u) b( u) = u u ( α )( u) O base of the cubc α-berste bass fuctos the α-berste bass fuctos of degree ( 4) ca be geerated accordg to the recurso property of the classcal Berste bass fuctos The the α-berste bass fuctos of degree ( 4) ca be defed as follows Defto The followg fuctos of u are called the α-berste bass fuctos of degree ( 4) b ( u) = ( u) b ( u) ub ( u) (9) where u < α = ad settg b ( u) = b ( u) By sple deducto expresso of the α-berste bass fuctos of degree ( 4) ca be got fro Eq (8) ad Eq (9) For exaple whe = 4 the quartc α-berste bass fuctos ca be expressed as follows b4 ( u) = ( u) ( u) ( α ) u b4( u) = u( u) ( )( u) ( α) u b4 ( u) = 6α u ( u) () b4 ( u) = u ( u) ( ) u ( α)( u) b44 ( u) = u u ( α )( u) where u < α ropertes of the bass fuctos For the sake of coveece the α-berste bass fuctos of degree ( ) are called α-berste bass fuctos for short the followg dscusso Fro the costructo process of the α-berste bass fuctos soe propertes of the bass fuctos ca be obtaed as follows Theore The α-berste bass fuctos defed as Eq (8) ad Eq (9) have the followg propertes (a) No-egatvty: b ( u) ( = ) (b) Noralzato: b ( u) = (c) Syetry: b ( u) = b ( u) ( = ) (d) ropertes at the edpots: = = b () = b () = ( α) = b () = α = α = b () = ( α) = roof Matheatcal ducto s used to prove ths theore (a) Whe = fro Eq (8) b ( u) ( = ) follow obvously because u ad α Suppose that the α-berste bass fuctos satsfy o-egatve for = Whe = fro Eq (9) the b ( u) = ( u) b ( u) ub ( u) ( = ) By the ductve hypothess ad the fact that u u t s obvously that b ( u) E-ISSN: Volue 4 8

4 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L ( = ) (b) Whe = fro Eq (7) t s easy to coclude that = b ( u) Suppose that the α-berste bass fuctos satsfy oralzed for = Whe = by the ductve hypothess ad Eq (9) the b ( u) = ( u) b ( u) = = ( ) = = u b u u u (c) Whe = the cubc α-berste bass fuctos satsfy syetry ca be obtaed by sple deducto fro Eq (8) Suppose that the α-berste bass fuctos are syetrcal for = Whe = by the ductve hypothess ad Eq (9) the b ( u) = ub ( u) ( u) b ( u) = ub ( u) ( u) b ( u) = b ( u) (d) Whe = fro Eq (7) the b ( u) = 6(α ) u ( α) u b ( u) = αu 9αu () b ( u) = 6αu 9αu b ( u) = 6( α) u ( ) u By sple deducto fro Eq (7) ad Eq () the = = b () = b () = α = α = b () = α = b () = α = = = Suppose that the α-berste bass fuctos hold the propertes at the edpots for = Whe = by the ductve hypothess ad Eq (9) the = b () = b () = () = b () = b () = Fro Eq (9) the b () = b () b () b () () b () = b () b () b () (4) By the ductve hypothess ad Eq () the followg coclusos ca be got fro Eq () () If = the b () = b () b () = α = (( ) α) () If = the b () = b () b () b () = α = ( ) () If the b () = Slarly the followg coclusos ca be got fro Eq (4) () If = the b () = ( ) () If = the b () = (( ) α) () If the () = b Theore shows that the α-berste bass fuctos have ost propertes of the classcal Berste bass fuctos artcularly the α-berste bass fuctos would degeerate to the classcal Berste bass fuctos for α = Hece the α-berste bass fuctos are sple extesos of the classcal Berste bass fuctos α-bézer curve Defto ad propertes of the curve O base of α-berste bass fuctos the correspodg Bézer-lke curve ca be aturally defed as follows Defto Gve cotrol pots ( = ) R or R for u < α r( u) = b ( u) (5) = s called α-bézer curve where b ( u ) ( = ; ) are the α-berste bass fuctos expressed as Eq (8) ad Eq (9) Fro Theore the α-bézer curve defed as Eq (5) has the followg propertes (a) Teral propertes: Fro the propertes at the edpots of the α-berste bass fuctos the r () = r() = ; r () = ( α)( ) r () = ( α)( ) Hece the α-bézer curve terpolates the frst ad the ed cotrol pots ad taget to the frst ad the ed edges of the cotrol polygo E-ISSN: Volue 4 8

5 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L (b) Syetry: Fro the syetry of the α-berste bass fuctos the r ( u; ) = b ( u) = = b ( u) = b ( u) j j = j= r( u; ) = Hece both ( = ) ad ( = ) defe the sae α-bézer curve a dfferet paraeterzato for the sae shape paraeter α ( < α ) (c) Geoetrc varat ad affe varace: Due to paraetrc for of the α-bézer curve the locato ad shape of the curve deped oly o the cotrol pots ( = ) ad the shape paraeter α regardless of the choce of coordate syste e the shape of the curve wll keep uchaged after rotato ad coordate traslato I addto after pleetg affe trasforato to the cotrol pots the ew curve wll correspod to the sae affe trasforato curve (d) Covex hull property: Because the α-berste bass fuctos are oegatve ad su to oe the α-bézer curve les sde ts cotrol polygos spa by the cotrol pots ( = ) It s clear that the α-bézer curve has ost propertes of the classcal Bézer curve artcularly the α-bézer curve would degeerate to the classcal Bézer curve for α = Hece the α-bézer curve s a exteso of the classcal Bézer curve Reark I cotrast wth soe exstg slar odels the α-bézer curve preseted ths work has the followg characterstc (a) I cotrast wth the o-polyoal Bézer-lke curves wth shape paraeters (such as [-6]) the α-bézer curve s polyoal Hece structure of the α-bézer curve s spler tha those odels based o the o-polyoal bass fuctos (b) I cotrast wth the hgh-degree Bézer-lke curves wth shape paraeters (such as [7-]) the α-bézer curve s stll a polyoal of the sae degree Hece forula coplexty of the α-bézer curve s spler tha those odels costructed by creasg the degree of the Berste bass fuctos (c) I cotract wth the Bézer-lke curve wth ultple shape paraeters of the sae degree [] the α-bézer curve satsfes strct syetry that the classcal Bézer curve has Hece the α-bézer curve s ore sutable practcal egeerg tha the odels that dd ot satsfy strct syetry (d) I cotract wth the Bézer-lke curve of degree wth - shape paraeters [] value rage of the shape paraeter of the α-bézer curve s fxed whch akes ore use-fredly operato for users Effects of the shape paraeter o the curve For fxed cotrol pots ( = ) shape of the classcal Bézer curve caot be chaged whle shape of the α-bézer curve ca be adjusted by alterg value of the shape paraeter α ( < α ) I order to dscuss effects of the shape paraeter α o the α-bézer curve a lea s gve frstly as follows Lea The α-berste bass fuctos defed as Eq (8) ad Eq (9) satsfy that 4 (a) b = b = ( ) (b) There exst costats c such that 4 b = b c = ( ) (c) There exst costats k such that 4 b = b k = ( = ; 4) roof Fro the syetry of the α-berste bass fuctos the b = b ( ) b b = ( ) b b = ( = ; 4) Hece oly the other half of every equato eeded to be proved Matheatcal ducto s used to prove (a) Whe = fro Eq (8) t s easy to 4 coclude that b = Suppose that 8 E-ISSN: Volue 4 8

6 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L 4 b = for = Whe = by the ductve hypothess ad Eq (9) the 4 b = b = (b) Whe = fro Eq (8) t s easy to 4 coclude that b = c where 8 c = For = suppose that there exst 4 costats c such that b c = Whe = by the ductve hypothess ad Eq (9) the b = b b 4 4 = c 4 = c where c = c (c) Whe = 4 the = fro Eq () t ca coclude that 4 b4 = α = k where k 4 = For = suppose that there exst costats k such that 4 b k = ( = ) Whe = fro Eq (9) the b = b b ( = ) By the ductve hypothess ad the results have bee proved (b) the 4 b k α = 4 4 c k = where k = k c O the base of Lea effects of the shape paraeter α o α-bézer curve approachg to ts polygo ca be show as follows Theore For fxed cotrol pots ( = ) suppose ( = ) le o the sae sde of edge The α-bézer curve defed as Eq (5) approaches closer to ts cotrol polygo as the shape paraeter α creases roof Whe ( = ) le o the sae sde of edge let * ( )( ) = (6) Fro Eq (5) ad Lea the * r = b b b = b b 4 = c k c (7) = where < α c ad k are costats Takg the or Eq (7) the r * 4 = c k c (8) = Whe cotrol pots = ( = ) are fxed c k c Eq (8) would 4 keep uchaged Sce decreases as α creases the α-bézer curve defed as Eq (5) approaches closer to ts cotrol polygo wth the crease of α Reark For ease of uderstadg set ( j) ( j) () ( j ) = = the Eq (6) ca be rewrtte as follows ( ) ( ) * = (9) * By sple deducto relatos betwee ad ( = ) for α-bézer curve of degree ( ) ca be got fro Eq (9) For exaples * whe = = ; whe = 4 * = Whe cotrol pots are fxed effects of the shape paraeter α o cubc α-bézer curve ad quartc α-bézer curve s show Fg ad Fg E-ISSN: Volue 4 8

7 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L respectvely where value of the shape paraeter α s set for α = 9 respectvely fro outsde to sde Fg Cubc α-bézer curve for dfferet α Fg Quartc α-bézer curve for dfferet α Splcg of the curve Gve two segets of adjacet α-bézer curves r () t = b () t ad r () t = b j() t j j j j= j= where the forer ad the latter s a α-bézer curve of degree ( ) ad degree ( ) respectvely The shape paraeter of r () t ad r () t s α ad α respectvely Geerally r () t ad r () t would satsfy G cotuous f r () = r () () r () = δ r () where δ s a gve costat Furtherore r () t ad r () t would satsfy C cotuous f δ = Eq () The the splcg codtos of r () t ad r () t satsfyg G ad C cotuous ca be show as follows Theore Two segets of adjacet α-bézer curves r () t ad r () t would satsfy G cotuous f = ad are collear Furtherore r () t ad r () t would satsfy C cotuous f = λ where λ s a gve costat roof By the teral propertes of the α-bézer curve the r () = r () = ( α)( ) () r () = r () = ( α )( ) If = ad are collear there would ext a costat λ such that = λ( ) () Fro Eq () ad Eq () the r () = r () () r () = λ r () Eq () shows that the two adjacet curves satsfy G cotuous Furtherore f = λ vz λ = Eq () would be rewrtte as follows r () = r () (4) r () = r () Eq (4) shows that the two adjacet curves satsfy C cotuous Suppose a whole G cotuous curve s splced by a uber of α-bézer curves of dfferet degree Fro Theore oly shape of the th curve seget would be locally adjusted f alterg value of the shape paraeter α whle shapes of the other curve segets would keep uchaged Whe the shape paraeters of all the curve segets are set for α = α the shape of the whole G cotuous curve ca be globally adjusted by alterg value of the shape paraeter α For choosg proper cotrol pots local adjustet of the shape paraeter α o a whole G cotuous curve splced by three segets of α-bézer curves s show Fg where the frst E-ISSN: Volue 4 8

8 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L ad the thrd curve segets are quartc ad the secod curve seget s cubc I Fg the shape paraeters of the frst ad the thrd curve segets are set for α = α = 5 the shape paraeters of the secod curve segets s set for α = 6 (dotted les) α = 8 (sold les) ad α = (dashed les) respectvely be globally adjusted For choosg proper cotrol pots global adjustet of the shape paraeter α = α ( = ) o a whole C cotuous curve splced by three segets of cubc α-bézer curves s show Fg 5 where the shape paraeter α s set for α = 6 (dotted les) α = 8 (sold les) ad α = (dashed les) respectvely Fg Local adjustet of a whole G cotuous curve For the sae cotrol pots Fg global adjustet of the shape paraeter α = α ( = ) o the whole G cotuous curve s show Fg 4 where the shape paraeter α s set for α = (dotted les) α = 6 (sold les) ad α = 9 (dashed les) respectvely Fg 4 Global adjustet of a whole G cotuous curve Suppose a whole C cotuous curve s splced by a uber of α-bézer curves of dfferet degree Fro Theore for the whole curve satsfyg C cotuous values of the shape paraeters of the other curve segets would chage f alterg value of the shape paraeter of the th curve seget The shape of the whole C cotuous curve would Fg 5 Global adjustet of a whole C cotuous curve 4 Cocluso The Berste bass fuctos wth a shape paraeter preseted ths paper have the sae propertes to those of the classcal Berste bass fuctos The Bézer-lke curve defed by the troduced bass fuctos ot oly has ost propertes of the classcal Bézer curve but also ca be easly adjusted by alterg value of the shape paraeter I costruct wth other slar odels the Bézer-lke curve preseted ths paper s stll a polyoal odel of the sae degree Hece t has spler structure Because there s early o dfferece structure betwee the proposed Bézer-lke curve ad the classcal Bézer curve t s o dffcult to adopt the proposed Bézer-lke curve to a CAD/CAM syste that already uses the classcal Bézer curve For practcal applcatos of the proposed Bézer-lke curve geoetrc odelg t s clear that soe specal algorths eed to be establshed Furtherore the correspodg Bézer-lke surface also eeds to be dscussed Soe terestg results ths area wll be preseted the followg study E-ISSN: Volue 4 8

9 WSEAS TRANSACTIONS o SIGNAL ROCESSING Jucheg L Ackowledgets Ths work was supported by the Scetfc Research Fud of Hua rovcal Educato Departet of Cha uder the grat uber 4B99 The author s also very grateful to the Hua provcal key costructo dscple Coputer Applcato Techology of Hua Isttute of Huates Scece ad Techology of Cha Refereces [] J Zhag C-Bézer curves ad surfaces Graphcal Models ad Iage rocessg vol 6 o pp [] Q Che ad G Wag A class of Bézer-lke curves Coputer Aded Geoetrc Desg vol o pp 9 9 [] J Zhag F Krause ad H Zhag Ufyg C-curves ad H-curves by extedg the calculato to coplex ubers Coputer Aded Geoetrc Desg vol o 9 pp [4] X Ha Y Ma ad X Huag The cubc trgooetrc Bézer curve wth two shape paraeters Appled Matheatcs Letters vol o pp 6-9 [5] U Bashr M Abbsa J MAl The G ad C ratoal quadratc trgooetrc Bézer curve wth two shape paraeters wth applcatos Appled Matheatcs ad Coputato vol 9 o pp 8-97 [6] J L A class of cubc trgooetrc Bézer curve wth a shape paraeter Joural of Iforato ad Coputatoal Scece vol o pp 7-78 [7] W Wag ad G Wag Bézer curves wth shape paraeters Joural of Zhejag Uversty SCIENCE A vol 6 o 6 pp [8] X Ha Y Ma ad X Huag A ovel geeralzato of Bézer curve ad surface Joural of Coputatoal ad Appled Matheatcs vol 7 o pp [9] L Yag ad X Zeg Bézer curves ad surfaces wth shape paraeters Iteratoal Joural of Coputer Matheatcs vol 86 o 7 pp [] L Ya ad Q Lag A exteso of the Bézer odel Appled Matheatcs ad Coputato vol 8 o 6 pp [] J Che ad G Wag A ew type of the geeralzed Bézer curves Appled Matheatcs: A Joural of Chese Uverstes vol 6 o pp [] T Xag Z Lu W Wag ad Jag A ovel exteso of Bézer curves ad surfaces of the sae degree Joural of Iforato ad Coputatoal Scece vol 7 o pp 8-89 [] X Q G Hu N Zhag X She ad Y Yag A ovel exteso to the polyoal bass fuctos descrbg Bezer curves ad surfaces of degree wth ultple shape paraeters Appled Matheatcs ad Coputato vol pp -6 [4] G Far Curves ad Surfaces for Coputer-Aded Geoetrc Desg (4ed) Elsever Scece & Techology Books Marylad 997 E-ISSN: Volue 4 8