On Local Fairing Algorithm for Cubic B-spline with the Second Discrete Curvature

Transcription

1 rd Iteratoal Coferece o Mechatrocs ad Idustral Iforatcs (ICMII 5 O Local Farg Algorth for Cubc B-sple wth the Secod Dscrete Curvature Cheg Cheg a Yu Desheg* b School of Matheatcs ad Iforato Scece Nachag Hagkog Uversty Nachag 6 Cha Matheatcal Secto Huage Nuber Hgh School Huage Couty Hube Provce 6 Cha a yxuxagfa@6co b 6co Keywords: dscrete curvature dfferetal farg cubc B-sple Abstract I ths paper a ew dscrete curvature-the secod dscrete curvature s troduced It s obtaed that the curvatures of a cubc B-sple curve at ts jog pots are proportoal to the secod dscrete curvatures of ts correspodg cotrol pots about the ode vector ad the secod dscrete curvature has slar propertes but be ore accurate tha dscrete curvature The farg algorth for cubc B-sple based o the secod dscrete curvature s gve Wth ths algorth curves are fared through adjustg the secod dscrete curvature of the correspodg cotrol pots drectly thus the farg process s ore cocse ad has stroger geoetrcal tuto It s showed that the algorth ca get better farg effects through experet exaples Itroducto Farg of curves ad surfaces s a portat topc of Coputer Aded Desg vestgato It has portat theoretcal ad practcal value A lot of researches have bee ade ad soe effectve farg algorths are obtaed ths feld The algorths ca aly be dvded to three categores: Curvature-based farg algorths Eergy-based farg algorths ad Curvature ad eergy-cobed farg algorths Curvature-based farg algorths such as Polakoff's autoatc farg algorth [] MCV farg algorth [] L's target curvature algorth [] ad Gle's curvature farg ethod [4] aly adjust the sple cotrol pots of curves ad surfaces (data pots to esure that ther approprate curvature aps are cotuous ad ootoous; Eergy-based farg algorths such as least square farg ethod [5] eergy farg ethod [6] ad Robet's shape preservg terpolato farg algorth [7] geerally based o physcal deforato eergy fucto fd the sallest physcal deforato eergy curves ad surfaces uder soe geoetrc ad o-geoetrc costrats; Curvature ad eergy-cobed farg algorths such as local eergy farg algorth [8] bass sple farg algorth [9] Zhag's u eergy farg algorth [] detere the areas to be fared aly through curvature at frst ad the far the wth eergy zg ethods There are soe specal farg algorths but to our kowledge researches o dscrete curvature-based farg algorths [] through adjustg the geoetrc postos of cotrol pots drectly are stll rare Sce B-sple curves have the propertes that they ca be descrbed atheatcally wth cotrol pots ther shapes ca be reflected by ther cotrol polygos ad ther shape odfcato ca be acheved by adjustg cotrol pots so f we ca far the curves ad surfaces through odfyg cotrol pots of B-sple ther farg process wll be ore drect To ths as based o the studes of relato betwee the curvatures at the odes of B- sple ad dscrete curvatures at ther correspodg cotrol pots ths paper troduce a ew dscrete curvature - the secod dscrete curvature It s obtaed that the curve curvatures at odes of B-sple curves ad the secod dscrete curvatures at the correspodg cotrol pots are proportoal ad the farg algorth based o the secod dscrete curvature for B-sple curve s gve Thus through adjustg the secod dscrete curvatures at cotrol pots properly the farg of B-sple curve segets at the edpots ad the purposes of farg the 5 The authors - Publshed by Atlats Press 464

2 B-sple curves ca be acheved So the farg processes are ore cocse ad have ore geoetrc tutos ad the results of [] are geerated Cocept ad property of the secod dscrete curvature Cocept of dscrete curvature Gve a sequece of pots plaar Q ( x y( L whch do t cocde Let L Q Q be the dstace betwee Q ad Q the the dscrete curvature at pot Q s defed as [] xy &&& &&& xy C ( / ( x& y& f b f b x x y y f x x b x x x& x& y& y& where x& y& ; x& x& ;&& x && y are the frst ad the secod L L L L L L L L L L dervatves of coordates to the le seget dstaces at potq Theore Let Q( x y( L be a sequece of plaar pots whch do t cocde M Q Q be the dstace betwee Q ad Q DQ QQ be the drected area of tragle Q QQ [4] the the dscrete curvature at pot Q s ( L L DQ QQ C ( LL M Proof Forula ( s obtaed by substtutg the frst ad the secod dervatves of coordates to the le seget dstaces at pot Q to forula ( ad the splfyg Reark For the two ed pots Q Q oly add two vertces Q Q Q ad Q Q Q ad substtutg DQ QQ ad D Q QQ as the correspodg drected areas of the tragles the the dscrete curvatures of the two boudary pots Q Q ca also be calculated by forula ( respectvely Basc cocept of the secod dscrete curvature ( ( Let QQ LQ be a plaar polygo ad deote Q j Q j ( j L Cuttg the corer by ( ( dvdg Qj Q j wth proporto µ λ ( µ λ > µ λ < the two ew vertces at left ad rght ( sdes of Q j are obtaed [5] ( ( ( Qj ( λj Qj λjqj j L ( ( ( Qj µ jqj ( µ j Qj ( ( ( ( ( ( ( ( ( where Q Q Q Q f Γ s closed ad Q Q Q Q f ( Γ s ope Coectg these ew ( ( ( ( ( ( ( vertces Q Q L Q successvely we obta a polygo Γ Q Q LQ the Γ ( s called the geeratg polygo fro cuttg Γ ( s corers ( Defto Let Γ QQ LQ be the cotrol polygo of a cubc B-sple curve cuttg Γ ( s u corers wth proportos u u u λ µ the u u u 4 u K DQ λ ( ( µ QQ Q Q Q Q ( s called the secod curvature of the cubc B-sple at vertces Q As a specal case whe Γ ( s the cotrol polygo of a cubc ufor B-sple curve e the dstaces of all ode tervals u u uare equal to a costat the the secod curvature of the cubc B-sple at vertces Q s K 7DQ M QQ Defto The frst ad the secod dffereces at vertces Q of a cubc B-sple curve are defed respectvely as K K K K K ; 465

3 D D D τk ( τ K K L whereτ L ( L L Basc propertes of the secod dscrete curvature Let Q% λq ( λ Q Q% ( µ Q µ Q σ L L l Q Q% l Q Q% be the dstaces betwee Qad %Q; Qad %Q respectvely α < l l > be the agle betwee les l ad l The we have the followg: Proposto The relato betwee the dscrete curvature ad the secod dscrete curvature of a sequece of pots Q( L o plae R whch do t cocde s C K L L Q% Q % LL M ( Proposto The secod curvatures at two overlappg cotrol pots ad the curvatures at the correspodg pot o B-sple curve are zero but the dscrete curvatures at those pots do ot exst I fact f two cotrol pots cocde aelyq Q the the three pots Q Q Q are collear ad %Q Q thus D % % Q λ QQ Q Q Q Q Therefore the establshet of the above s cocluded Proposto Let L L L L α < α the the relato of the two secod dscrete curvatures at cotrol pot Q s K > K I fact DQ LL s α QQ K Q% Q% ( l l ll cos α / Proposto 4 Let L < L L < L α α ad L L dscrete curvatures at cotrol pot Q s K > K I fact ths case we have l λ L l µ L thus M L L cos LL α % % Q Q l l ll cosα Because Q QQ Q QQ we have σ σ λ λ µ µ Thus M M % % % % Q Q Q Q So D L L sα M K % % Q Q Q % Q % Q QQ M M M LL sα > K ( l l ll cos / α L L the the relato of the two secod σ σcosα cos λ µσ λµσ α L L s D > K M M M α M Q QQ Q% Q% Q% Q % Proposto 5 The relato of the frst ad the secod dffereces of the secod dscrete curvatures at cotrol pot Q s LL K K D ( L L L L Fro ths t s coclude that D shows that the rato of the frst dfferece of the frst dscrete curvature ad the legth of the chord vares uforly; the saller D ore evely the chages of the rato of the frst dfferece betwee the frst dscrete curvature ad the legth of the chord 466

4 Soothg prcple based o the secod dscrete curvatures Relato betwee the curvatures at the edpots of cubc B-sple curve segets ad the secod dscrete curvatures of ther cotrol pots Let pu N u u u ( Q [ ] be a cubc B-sple curve detered by cotrol pots Q Q L Q Wheu [ u u ]( L the curve seget s detered by four cotrol potsq QQ Q ad deoted by p ( u( L We have the followg Theore Let κ be the curvature at the edpot p( u of a cubc B-sple curve seget p ( u( L K be the secod dscrete curvature at ts correspodg cotrol vertces Q the ( u u( u 4 u K κ (4 4( u u Proof The curvature at the edpot p( u of the cubc B-sple curve seget s p ( u p ( u κ p ( u By the propertes of cubc o-ufor B-sple curves we kow the taget vector ad the secod dervatve vector at the pot p ( u of p ( u are p ( u u ( u ( ( u u ( u u Q Q ( u u ( Q Q ( u u ( u u u u u u u 4 u 4 p 6 ( u ( ( u u( u u 6 ( Q Q ( u 4 u ( u u Thus p ( u u u u u 6DQ QQ ( Q Q ( Q Q p ( u p ( u ( u u ( u u ( u u 4 So 4( u u DQ QQ κ (5 ( u u( u 4 u λ ( Q Q µ ( Q Q Therefore forula (4 s followed edately fro forulae ( ad (5 Corollary Let p ( u( L be a ufor cubc B-sple curve seget the K 7κ 6 (6 I fact forula (6 s edately yelded by settg u u uto be a costat forula (4 ad the splfyg Soothg crtero based o the secod dscrete curvature I curve odelg wth B-sple the curve s usually costructed ftely dfferetable wth each curve seget but the dfferetablty at a ode depeds o repeat uber of the ode Thus the farg process of B-sple curves s the farg process at the edpots of B-sple curve segets correspodg to the ode vector; ad we eed to cosder the curvatures at edpots oly I geeral people ted to take the curvatures as a drectory fucto of curve soothess Itutvely f the curvature of a curve chages slowly the the curve s sooth [6] Coversely f the curvature chages rapdly the followg two case wll appear: (a The bedg drecto of the curve chages the the sg of curvature chages aely: there s a extra fecto pot; (b The sg of curvature does t chage but the curvature values chages rapdly wth a certa rage the extra bup curve segets appear The curves are certaly ot sooth f the above two cases appear Based o the above aalyss ad defto of the secod dscrete curvature we beleve that the curve s sooth f the uber of the curve fecto pots ad the uber of curve segets are al uder the crcustace to esure the desg requreets The we ca detere bad pots ad ther types by the followg crtera 467

5 ( To esure the u uber of curve fecto pots (preferably zero we refer the pot Q satsfyg the codtos KK ad K K > as a bad pot of the frst kd I ths case the sg of curvature at the ode u chages ( As a extree curvature value correspods to a cocave (covex curve seget therefore we refer the pot Q at whch the sg of frst order dfferece chages aely satsfyg codto K K < as a bad pot of the secod kd I ths case the curvature terval[ u u ] s o-ootoc ad thus the reaches a local extree at the ode u ( Because B-sple curves have strog covex hull property we refer the pot Q ot satsfyg covexty codto K > [( τ K τk ] as a bad pot of the thrd kd uder the crcustace to esure costat curvature sg Adjustg of bad pots I ths part we deote a ax{ u u} b { u 4 u } ad cosder the local optzato algorth whch aly ze the eergy of the curve uder a acceptable chage of the shape of the curve thus the lear cobato of the eergy fucto ad error fucto of the curve: α G ( δ ( p ( u du δ β ( Q Q (7 s used as the objectve fucto where whe δ [] Adjustg of a cotrol pot Suppose Q s a sgle bad pot the sequece of cotrol pots the adjustg ts posto wll lead to a shape chage of the curve seget defed o terval[ ab ] I other words the adjustet of Q relate wth at ost the sx potsq- Q- Q- Q Q Q By forula (6 the pot after adjustg be where b Nk unk u du b ( N k ( u du a Q δq ( δ δ Q (8 ( ( ( a δ [] δ ; ax{ } { } Adjustg of two cotrol pots Suppose Q ad Q are two cosecutve bad pots the sequece of the cotrol pots adjustg ther postos wll lead to a shape chage of the curve seget defed o terval[ ab ] By forula (6 the pots after adjustg satsfy the followg lear equatos ( δ δ % δq ( δ δ Q Q ( δ δ % Q (9 δq ( δ δ Q where b Nk un jk u du b ( N jk ( u du a ( ( ( j a δ [] δ ; ax{ } { 4 } Obvously the rght sdes of equatos (8 ad (9 are the weghted averages of the orgal cotrol pots ad δ [] s the odfyg factor thus the greater δ the saller the offset of cotrol pots Reark We fd that adjustg three adjacet cotrol pots wll lead the ew curve to obvous bas wth the shape of the orgal curve so ths paper oly cosder the algorth of the case to adjust two adjacet cotrol pots at ost 468

6 Cotrollg of errors Theore Let Q be a bad cotrol pot of the orgal curve Q be the cotrol pot of the curve after farg ad Q Q < ε the p( u p ( u < ε Proof By the defto ad propertes of B-sple curves we have p( u p( u Q N ( u Q N ( u N ( u( Q Q N ( u Q Q < N ( u ε ε N ( u ε I order to cotrol the error betwee the orgal curve ad the soothed curve to ay gveε > we adopt f Q s a sgle bad pot; ad adopt δ ax{ ε Q δ Q } δ ax{ ε Q δ Q ε Q δ Q } f Q s the frst bad pot two cosecutve bad pots The the error betwee the orgal curve ad the soothed curve ca be cotrolled the rage ofε I fact Q Q δq ( δ δ Q Q ( δ ( Q δ Q δ Q δ Q < ε Farg algorth dκ dκ I ay farg algorths Sapds' defto z ( u ( u ad ξ z are used as partal ds ds ad whole curve farg dcator fucto respectvely Slarly we defe z D as a dscrete local curve farg fucto ad ξ D as whole dscrete curve farg fucto Here D s the secod-order dfferece at cotrol pot Q the physcal eag of ts absolute value D s the exteral pressure of ro to the sple of wood Thus the larger (saller D the easer (ore dffcult the curve deforato at the correspodg pot ad the curve s ot sooth (sooth We gve the followg farg algorth detal: Step Eter the sequece of cotrol pots Q Q L Q out; Step Detere bad pots ad ther types ad calculate local farg fucto value z at each bad pot ad the overall farg dcator fucto valueξ of the curve; Step Order z decreasgly: z z L ad the odfy the successvely ths order; Step 4 If j j > oly adjust pot Q ; Otherwse adjust two pots Q j ad Q ; j j Step5 Output a seres of ew cotrol potsqq % % L Q % ad calculate the overall farg fucto value ξ after adjust If ξ decreases forward to step 6; otherwse retur step ; Step 6 Geerate a ew curve wth ew cotrol potsqq % % L Q % Experetal exaples I ths part we choose followg two coo exaples to verfy the valdty of the above algorth Exaple Take a seres of cotrol pots ad odes we obta a rabbt curve wth the costructg ethod of o-ufor cubc B-sple curves Obvously t s ot s sooth (Fg The curve s fared by the ethod proposed Secto4 ad the rabbt curve after farg s show Fg By coparg Fg ad Fg4 we fd that ot oly the shape of the curve after farg s slar to that of the orgal oe s but also ts dscrete curvature ad the secod dscrete curvature are uch saller tha those of the orgal oe s Hece the curve obtaed by our farg ethod s soother 469

7 Fg Rabbt curve before farg wheε Fg Rabbt carve after farg wheε C K Fg The dscrete curvature graph Fg4 The secod dscrete curvature graph of 88 cotrol pots after farg of 88 cotrol pots after farg Exaple Take a seres of cotrol pots ad odes we obta a S curve wth the costructg ethod of o-ufor cubc B-sple curves Fg5 to Fg are the curvature coparso graphs of orgal curves ad the curves whch are fared by the ethod of secto 4 whereε takes 5 respectvely We fd that the greater ε the ore getle the curvature chage of the curve thus the better the curvature ootocty ad soother the curve but the greater shape chage of the curve 5 5 Fg5 Coparg graph of the curve of S type Fg6 Curvature Coparg graph of the curve of S before ad after farg wheε type before ad after farg wheε takesε Fg7 Coparg graph of the curve of S Fg8 Curvature Coparg graph of the curve of type before ad after farg wheε S type before ad after farg wheε 4 47

8 5 5 4 Fg9 Coparg graph of the curve of S Fg Curvature Coparg graph of the curve of type before ad after farg whe ε 5 S type before ad after farg wheε 5 Coclusos There are ay farg ethods for B-sple curves I ths paper the cocept of the secod dscrete curvature s put forward the propertes of the secod dscrete curvature ad ts dffereces are vestgated The farg algorth based o the secod dscrete curvature for cubc B-sple curves s gve Ths farg algorth has uque advatages copared wth tradtoal oes Frstly ths ethod s sple ad drect just by odfyg bad pots the cotrol pots of o-ufor cubc B-sple curves the farg purpose ca be acheved; Secodly the farg algorth has stroger geoetrc tuto As the secod dscrete curvature s a expresso of the tragle s area ad ts edge legths t s a geoetrc varat so ts geoetrc tuto s stroger I addto sce B-sple surfaces are tesor product of B-sple curves the results of B-sple curves ca be exteded to B-sple surfaces Refereces [] J F Polakoff Y K Wog P D Thoas A autoated curve farg algorth for cubc B-sple curves [J] Joural of Coputatoal ad Appled Matheatcs 999 :7 85 [] Yul Wag Bgya Zhao Luzhou Zhag Jachua Xu Kachag Wag Shuchu Wag Desgg far curves usg ootoe curvature peces [J] Coputer Aded Geoetrc Desg 4 : 5 57 [] Wesh L Shuhog X Ja J Zhao G Target curvature drve farg algorth for plaar cubc B-sple curves [J] Coputer Aded Desg 4 : [4] Gle Muleux Sebasta TRobso Farg pots sets usg curvature [J] Coputer Aded Geoetrc Desg 7 9: 7 4 [5] Já Glasa Least-squares soothg of D dgtal curves [J] Real-Te Iagg 5 : [6] Xua Yag Guozhao Wag Plaar pot set farg ad fttg by arc sples [J] Coputer Aded Geoetrc Desg :5-4 [7] Robet J Reka Shape-preservg terpolato by far dscrete G space curves [J] Coputer Aded Geoetrc Desg 5 : [8] Eck M Hadefeld J Local eergy farg of B-sple curves [J] Coputg Suppleetu 995 :9-47 [9] X Yulog O Fttg ad Farg of Curves [J] Joural of Fuda Uversty Scece Edto 975 :- [] Cag Zhag Pfu Zhag Fuhua Cheg Farg sple curves ad surfaces by zg eergy [J] Coputer Aded Desg (:

9 [] Lu GH Wog YS Zhag YF Loh HT Adaptve farg of dgtzed pot data wth dscrete curvature Coputer Aded Desg 4:9- [] Yu Desheg Cheg Cheg O Local Farg Algorth of Cubc Ufor B-sple wth Dscrete Curvature Joual of Zhejag Uversty (Scece Edto 8(5:5-57 [] Carel E Cohe-Or D Warp-guded object-space orphg [J] The Vsual Coputer 997 : [4] Yu Desheg O Soe Theores of Drected Areas of Polygos [J] Joural of Gaa Teacher s College 999(:-4 [5] Wag Rehog L Chogju Zhu Chugag et al A Course Coputatoal Geoetry [M] Bejg: Scece Press 8 [6] Dog Guagchag Zhag Da Lu Zhcheg MA Lzhuag Curve farg [J] Progress Natural Scece 997 7(5: