Interactive Control of Planar Class A Bézier Curves using Logarithmic Curvature Graphs

Transcription

1 Computer-Aded Degn nd Applcton 8 CAD Soluton, LLC Interctve Control of Plnr Cl A Bézer Curve ung Logrthmc Curvture Grph Norm Yohd, Tomoyuk Hrw nd Tkfum Sto Nhon Unverty, norm@cmorg Nhon Unverty Tokyo Unverty of Agrculture nd Technology, txto@cctutcjp ABSTRACT We preent method for nterctvely drwng plnr cl A Bézer curve egment Frt, we preent method for nterctvely drwng typcl cl A Bézer curve egment y pecfyng three pont lke qudrtc Bézer curve egment We how tht the degree of typcl cl A Bézer curve egment elevted, the curve converge to logrthmc prl egment At the lmt of nfnte degree, the curve egment ecome logrthmc prl egment We lo preent method for drwng generl cl A Bézer curve egment y perturng the element of the typcl cl A mtrx o tht the endpont contrnt re tfed To ee the chrctertc of the generted curve, we propoe to ue logrthmc curvture grph Keywor: cl A Bézer curve, monotone curvture, logrthmc prl, logrthmc curvture grph INTRODUCTION For hghly ethetc hpe degn, uch the degn of cr ode, t very mportnt to ue ethetc curve key lne Uully, the curvture dtruton of uch curve re of prml mportnce nd the curvture of curve egment requred to e monotonclly vryng[] Hrd nt tht tghter retrcton hould e gven for ethetc curve ed on h nly of mny ethetc curve n rtfcl nd the nturl world[8,6] The retrcton the lnerty of logrthmc curvture grph (formerly clled logrthmc curvture htogrm) Mur derved the generl formul of log-ethetc curve (formerly clled Aethetc Curve) wth lner logrthmc curvture grph[] Yohd nd Sto clrfed the overll hpe of log-ethetc curve nd preented method for nterctvely drwng curve egment[4] Yohd nd Sto lo propoed Qu-log-ethetc curve (formerly clled Qu-Aethetc Curve[5]) tht pproxmtely repreent log-ethetc curve n rtonl cuc Bézer form In prctcl tuton, however, trct lnerty of the logrthmc curvture grph my e too retrctve Curve wth looer retrcton (pproxmte lnerty) or the curve tht cn control the lnerty of ther logrthmc curvture grph re dered In th pper, we nvetgte cl of curve wth monotonclly vryng curvture, whch re plnr cl A Bézer curve Cl A Bézer curve[4] re curve wth monotonclly vryng curvture nd toron propoed y Frn Though n nterctve generton method for typcl cl A Bézer curve (ee Secton ) known, the nterctve control method of generl cl A Bézer curve not known We frt preent method for nterctvely controllng typcl cl A Bézer curve egment y three pont lke qudrtc Bézer curve We how tht the degree of typcl cl A Bézer curve elevted, the curve converge to logrthmc prl A generl cl A Bézer curve egment nterctvely generted y perturng the cl A mtrx o tht endpont contrnt (the poton nd the tngent vector) re tfed To ee the chrctertc of the generted curve, we propoe to ue logrthmc curvture grph Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

2 The ret of th pper orgnzed follow Secton revew relevnt lterture Secton revew cl A Bézer curve Secton 4 decre logrthmc curvture grph nd ther chrctertc In Secton 5, we preent method for nterctvely controllng typcl cl A Bézer curve y pecfyng three pont lke qudrtc Bézer curve Then n Secton 6, we how tht the degree of typcl cl A Bézer curve elevted, the curve converge to logrthmc prl Secton 7 preent method for nterctvely controllng generl cl A Bézer curve Secton 8 how the generted cl A Bézer curve nd ther logrthmc curvture grph Fnlly, concluon re preented n Secton 9 RELATED WORKS Free-form curve nd urfce, uch Bézer or NURBS, re wdely ued n current CAD ytem From the vewpont of degnng ethetclly ppelng curve, uch the key lne of cr ode, uch free-form curve re very dffcult to mnpulte For ethetclly ppelng curve egment, degner my wnt to modfy the curve hpe under the contrnt of monotonclly vryng curvture However, the curve hpe of Bézer or NURBS re uully controlled y ther control pont (nd weght n ce of rtonl curve nd knot n ce of NURBS) It not n ey tk to plce or move control pont uch tht the curve egment h monotonclly vryng curvture Severl pper hve delt wth the generton of Bézer or B-plne curve wth monotonclly vryng curvture Spd nd Frey preented the necery nd uffcent condton of monotone curvture for qudrtc polynoml Bézer curve[] Frey nd Feld preented the condton for rtonl qudrtc curve[6] Dez nd Pper ued precomputed tle o tht polynoml cuc Bézer curve hve monotonclly vryng curvture[] Wng et l preented uffcent monotone curvture condton for polynoml Bézer nd B-plne curve of degree n [] More recently, Frn h propoed cl A Bézer curve[4] Co nd Wng preented the correct condton for cl A Bézer curve generted y ymmetrc mtrce[] The method for nterctvely drwng cl A Bézer curve egment not known except for the curve typcl curve Cl A Bézer curve re revewed n the next ecton Yohd nd Sto propoed log-ethetc curve[4], whch re formerly clled Aethetc Curve Log-ethetc curve re curve wth monotone curvture nd re ed on the lnerty of the logrthmc curvture grph Th ed on the oervton y Hrd[8,6] tht mny of ethetc curve n rtfcl nd the nturl world re curve whoe logrthmc curvture grph cn e pproxmted y trght lne We cll the lope of the trght lne α When α,,, ±, log-ethetc curve ecome the Clothod, the logrthmc prl, the crcle nvolute, nd the crcle, repectvely When α, we recently found tht the log-ethetc curve ecome the Nelen prl[7] Logethetc curve cn e condered the generlzton of thee curve A log-ethetc curve egment cn e nterctvely drwn y pecfyng three pont lke qudrtc Bézer curve when α pecfed[4] Qu-logethetc curve egment n rtonl cuc Bézer form hve lo een propoed[5] In Qu-log-ethetc curve egment, the lnerty of the trght lne n the logrthmc curvture grph pproxmtely preerved nd t monotoncty of the curvture gurnteed We ue logrthmc curvture grph for the nly of generted curve CLASS A BÉZIER CURVES In th ecton, we revew cl A Bézer curve[4] nd pont out tht typcl cl A Bézer curve of degree the pecl ce of Pythgoren hodogrph curve[5] Though cl A Bézer curve nclude pce curve wth monotone curvture nd toron, we only del wth plnr cl A Bézer curve n th pper Co nd Wng recently found tht the monotoncty of the curvture of cl A Bézer curve not proved[] Though they provded proof for ymmetrc mtrce, more nvetgton necery for the proof of generl mtrce However, the monotoncty of the curvture cn e ely checked y mplng the pont of curve nd computng the curvture The cl A condton, though proved only for ymmetrc mtrce, tll ueful for genertng the curve wth monotone curvture In our mplementton, we generte curve ung the cl A condton for ymmetrc mtrce nd the monotoncty of the curvture lwy checked y uffcently mplng the pont on the curve nd computng the curvture For plnr curve, we found tht the cl A condton for ymmetrc mtrce fl rrely for generl mtrce If correct cl A condton for generl mtrce found, our frmework for nterctvely controllng cl A Bézer curve wll tll work jut y replcng the cl A condton y the new one A plnr Bézer curve x () t of degree n defned y Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

3 n n () t B () t x () where B n () t Bernten polynoml nd re two-dmenonl control pont vector Let Δ + ( n ) e the forwrd dfference vector of the control pont Ung cl A mtrx M defned elow, cl A Bézer curve re defned y gvng the followng contrnt for the control pont vector M K, n () ( ) Cl A mtrx M mtrx uch tht for ny vector ( t) v + tmv v v nd for ny t [,], the followng reltonhp hol: () Th men tht the lne egment defned y v nd Mv for ny v doe not nterect the crcle whoe rdu except for t endpont See Fg For mtrx M to tfy Eqn() for ny vector v nd for ny t [,], the followng two condton mut hol () The ngle etween v nd Mv mut e le thn 9 degree () The mtrx M mut e n expnon mtrx tht trnform every pont of the unt phere to the outde of the phere To tfy condton (), Frn howed tht the followng mtrce mut hve nonnegtve egenvlue[4]: T T M + M I, M M I (4) where I the dentty mtrx To tfy condton (), the ngulr vlue σ, σ ( σ σ ) of M mut e greter thn or equl to To prove the monotoncty of the curvture, Co nd Wng gve the followng requrement for σ,σ of ymmetrc mtrce []: σ σ +, σ σ + (5) Thu, for ymmetrc mtrx M to e cl A, the mtrce of Eqn(4) mut hve nonnegtve egenvlue, the ngulr vlue σ,σ of M mut e greter thn or equl to nd Eqn(5) mut hold We cll thee condton for mtrx M cl A condton Agn, th cl A condton correct only for ymmetrc mtrce M We ue the condton for generl mtrce nd for gurnteeng the monotoncty of the curvture we mple the pont on the curve nd compute the curvture For plnr curve, we found tht the cl A condton fl rrely for generl mtrce Frn gve n exmple of cl A mtrce, whch we cll typcl cl A mtrce Typcl cl A Bézer curve generted y typcl cl A mtrce re the typcl curve of Mneur[], whch w npred y the curve of Hgh[9] If the mtrx M trnformton mtrx compoed of rotton y n ngle < π / followed y clng, nd the nequlty co > ( f > ) or co > ( f < ) (6) hol, the mtrx M ecome cl A The ltter nequlty( co > ) cn e derved y replcng wth n for, K,n In contrt, generl cl A mtrx mtrx tht tfe condton () nd () Generl cl A mtrce v M v v M v Fg An exmple of the cton of Cl A mtrx(left) nd NOT cl A mtrx (rght) π < co > Fg A Typcl Cl Bézer Curve Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

4 4 nclude typcl cl A mtrce, nd we cll the generted curve the generl cl A curve See Fg for n exmple of typcl cl A Bézer curve Note tht typcl cl A Bézer curve of degree the pecl ce of Pythgoren hodogrph curve[5] of degree If Pythgoren hodogrph curve of degree tfe Eqn (6), the curve ecome cl A Bézer curve For typcl cl A Bézer curve, Frn gve method for nterctve control y pecfyng two endpont nd ther tngent However, the method for nterctvely genertng generl cl A Bézer curve not known 4 LOGRARITHMIC CURVATURE GRAPHS Log-ethetc curve re curve whoe logrthmc curvture grph re repreented y trght lne Logrthmc curvture grph were orgnlly clled logrthmc curvture htogrm n [,4,5], whch re Mur nterpretton[] of orgnl logrthmc curvture htogrm of Hrd Note tht Hrd orgnl logrthmc curvture htogrm[8,6] were ctully htogrm Let ρ e the rdu of curvture nd e the rc length We ume tht ρ monotonclly ncreng wth repect to The fundmentl equton of log-ethetc curve[,4] log ρ α + c (4) dρ where α the lope of the trght lne n the logrthmc curvture grph nd c contnt Snce monotonclly ncreng functon of ρ, / dρ of log-ethetc curve re lwy potve However, / dρ of other curve my ecome negtve Thu we ue log( ρ / dρ ) for the vertcl x of the logrthmc curvture grph Tkng the olute vlue of / dρ men the reprmeterzton of the curve wth repect to For the horzontl x, mply ued From Eqn(4), we cn derve the functon of the curvture wth repect to the rc length See [4] for the detl of the dervton Eqn(4) fxe the curvture functon: Λ e f α (4) ( Λα + ) α otherwe where the curvture nd Λ dρ / t ρ Therefore, the lnerty of the logrthmc curvture grph men tht the curvture of the curve repreented y mple functon of the rc length We cn compute the logrthmc curvture grph for rtrry prmetrc curve, uch Bézer or NURBS curve The horzontl vlue cn e ely computed The vertcl vlue log( ρ / dρ ) cn e computed ung dρ d (4) nd ( ) d det( x&, x ) x& x& det( x&,&& x) x& && x 6 x& (44) Fg how three cuc Bézer curve egment wth ther curvture plot nd ther logrthmc curvture grph In the fgure (nd other fgure throughout the pper), whte crcle or lck crcle mrked to denote the trt pont or the end pont of egment, repectvely Fg () how curve egment wth not monotonclly vryng curvture Fg () how curve egment wth monotonclly vryng curvture ut not lner logrthmc curvture grph Fg (c) how curve egment wth monotonclly vryng curvture nd lner logrthmc curvture grph Logrthmc curvture grph cn e ued to ee the chrctertc of the curve Logrthmc curvture grph hve the followng chrctertc () Seeng the lnerty of the logrthmc curvture grph gve ome clue how the curve devted from logethetc curve In other wor, f the logrthmc curvture grph lmot lner, the curvture cn e pproxmtely repreented y mple functon of the rc length () If the horzontl vlue of the logrthmc curvture grph monotonclly ncreng or decreng, the curvture of the curve monotonclly vryng Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

5 5 (c) If the curvture of curve not monotonclly vryng or contnt, the curve nclude pont t d / At the / dρ ecome nfnte (d) pont, ( ) of the nflecton pont of curve nfnte ( ρ d / dρ ) undefned log of the nflecton pont ether nfnte or From the chrctertc (c) nd (d), the logrthmc curvture grph not good t curve wth not monotonclly vryng curvture or contnt curvture, or curve wth n nflecton pont However, we cn check the proxmty of the curve to log-ethetc curve The monotoncty of the curvture cn lo e checked log ρ () Curve wth NOT monotone curvture log ρ () Curve wth monotone curvture log ρ (c) Curve wth monotone curvture nd pproxmtely lner logrthmc curvture grph Fg Cuc Bézer curve(left), ther curvture plot(mddle) nd ther logrthmc curvture grph(rght) 5 INTERACTIVE CONTROL OF TYPICAL CLASS A BÉZIER CURVES For nterctvely controllng typcl cl A Bézer curve, we preent method for drwng typcl cl A Bézer curve of degree n y pecfyng three control pont lke qudrtc Bézer curve Typcl cl A mtrx M co n M (5) n co tht tfe Eqn(6) Let,, n We would lke to generte curve whoe endpont re nd nd the tngent vector t the endpont re prllel to nd, repectvely Snce nd n, we need to fnd,, K, n n Eqn(5) the ngle etween the vector nd the vector dvded y n See Fg 4 Let nd u We need to fnd nd uch tht e three pont vector for genertng typcl cl A Bézer curve egment of degree ( ) n j M u j ( ) (5) Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

6 6 We ue the optmzton proce to fnd nd uch tht n j ( ) M u ( ) f, (5) j ecome the zero vector Snce we cn compute the dervtve of Eqn(5) wth repect to nd, we cn ue the optmzton ung dervtve Note tht when n, Eqn(5) ecome of degree Thu, nd cn e unquely computed wthout ung the optmzton Once nd re computed, ll the other control pont vector re computed y j + M u j ( < m) (54) Δ Fg 4 Interctve control of typcl cl A Bézer curve 6 RELATIONSHIP WITH LOGARITHMIC SPIRALS In th ecton, we how tht typcl cl A Bézer curve re pproxmton to logrthmc prl More pecfclly, we how tht the degree of typcl cl A Bézer curve elevted, the curve converge to logrthmc prl A logrthmc prl knd of prl curve whch pper n nture, uch nutlu hell nd prl glxe Logrthmc prl re ncluded n log-ethetc curve the ce of α The equton of logrthmc prl whoe pole p t the orgn on the complex plne ( ) r e k LS (6) where the mgnry unt, r ( > ) nd k( cot( k) π ) re contnt The prl curve rotte round the pole p of the prl Fg 5() how n exmple of logrthmc prl The tngentl ngle the ngle etween x -x nd the vector trtng from the pole to LS ( ) The dtnce from the pole to the pont of the curve LS ( ) k LS r e ( ) We conder two pont on logrthmc prl whoe tngentl ngle re nd + d ( d > ), repectvely The rto of LS ( + d ) nd ( ) LS( + d ) k d e ( ) LS LS cont Therefore, f the tngentl ngle ncreed y d, the dtnce etween the pole to the curve ncreed y fctor k d of e Converely, for certn curve, f we could how tht the dtnce etween the pole nd the pont of the curve k d ncreed y fctor of e when the tngentl ngle ncreed y d, we cn y tht the curve logrthmc prl nce we cn ely fnd ll the prmeter of Eqn(6) We ue th property to how tht typcl cl A Bézer curve re pproxmton to logrthmc prl We re gven the control pont ( n) of cl A Bézer curve of degree n We wnt to how tht we cn unquely determne logrthmc prl tht goe through ll the control pont For control pont, +, + of cl A Bézer curve, we would lke to fnd the pole p of logrthmc prl uch tht the trngle p + nd (6) Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

7 7 p + + re mlr If we could fnd uch pole for rtrry, then + p / p + + / + hol nd there ext logrthmc prl tht goe through ll the control pont Here the clng fctor of the typcl cl A mtrx The pole ext on the Apollonu crcle, whch the locu of pont c whoe dtnce from + multple of t dtnce from () logrthmc prl () Apollonu crcle Fg 5 A logrthmc prl nd Bezer control pont wth Apollonu crcle To compute the pole p, we frt compute n Apollonu crcle c from nd + The Apollonu crcle c the locu of pont q uch tht + q / q, where + + / + In mlr mnner, we cn contruct Apollonu crcle c +, c+ from +, + nd +, +, repectvely From the crcle c nd c +, we cn contruct lne l tht goe through the nterecton pont of the two crcle Smlrly, we cn contruct nother lne l + from c+ nd c + The pole p cn e computed the nterecton of two lne l, l+ y checkng f the nterecton pont on the crcle c Let [ ] T e the frt control pont vector nd [ ] T x y u u x u y e the unt vector prllel to Then the other control pont of typcl cl A Bézer curve re gven y Eqn (54) For control pont,,,, the pole p cn e computed p [ p / / ] T x pw py pw (6) where p + + u + u co + u n p x x p y w y + x + y x + u y co ( ) ) x ( + u ) co + u n ) y x y y x We cn confrm tht the computed pole p on crcle c, thu on c + nd c + Therefore, for cuc typcl cl A Bézer curve, we cn compute the pole For cl A Bézer curve of degree 4, we cn ely how tht the trngle p nd p 4 re mlr y ung the reltonhp of p / p / 4 nd p p 4 By repetng th proce for cl A Bézer curve of degree n, we cn confrm tht the trngle p + nd p + + ( > ) re mlr Thu for ny typcl cl A Bézer curve of degree n, the pole p repreented y Eqn(6) For typcl cl A Bézer curve of degree n, we cn contruct logrthmc prl tht goe thorough ll the control pont Snce the control pont converge to the Bézer curve telf the degree elevted, typcl cl A Bézer curve converge to logrthmc prl Thu typcl cl A Bézer curve cn e condered n pproxmton to logrthmc prl 7 INTERACTIVE CONTROL OF GENERAL CLASS A BÉZIER CURVES Th ecton preent method for nterctvely controllng generl cl A Bézer curve To generte generl cl A Bézer curve, we trt from typcl cl A Bézer curve nd then pertur the element of the typcl cl A mtrx uch tht endpont contrnt re tfed (64) Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

8 8 We re gven three pont vector,, nd mtrx perturton prmeter δ We would lke to fnd generl cl A Bézer curve whoe endpont re, nd ther tngent vector re prllel to,, repectvely A generl cl A mtrx M G cn e otned y perturng typcl cl A mtrx M y δ : co + δ n + δ M G (7) n + δ co + δ If δ re ll nd M G tfe Eqn(6), M G ecome typcl cl A mtrx However, f t let one of δ not, the optmzton proce decred n Secton 5 doe not work nce chngng only nd T doe not tfy the endpont contrnt We dd the optmzton prmeter nd ue n optmzton functon n M G j j uch tht u ( ) nd the ngle φ formed y nd n n ecome See Fg 6 for φ We ue n optmzton proce to fnd,, uch tht n j (, ) u ( ) + φ f, j M G ecome If we do not drw the curve Whether poton of,, nd the vlue of δ, δ, δ M G doe not tfy the cl A condton or the curvture not verfed wth monotonclly vryng, M G generte curve wth monotonclly vryng curvture or not depen on the φ n Fg 6 Angle φ formed y nd n n Once,, re found, we cn ely generte the control pont of the cl A Bézer curve However, the generted curve not nvrnt under mlrty trnformton nce the ngle formed y u nd M G u dffer dependng on the drecton of u To mke the generted curve to e nvrnt under mlrty trnformton, we perform mlrty trnformton of, nd uch tht u ecome [ ] T nd ext n the frt qudrnt Then we perform the optmzton proce, compute the control pont nd trnform them ck 8 RESULTS Fg 7 how typcl cl A Bézer curve nd ther logrthmc curvture grph Fg 7 () how cuc typcl cl A Bézer curve nd t logrthmc curvture grph We ft lne to t logrthmc curvture grph ung the let qure method It lope 55 nd t vrnce 9 7 Fg 7 () nd (c) re typcl cl A Bézer curve of degree 6 nd, repectvely The lope of the ftted trght lne of the logrthmc curvture grph of Fg7() nd 6 (c) re 8 nd 97, nd ther vrnce re nd 4, repectvely Thee reult how how the curve egment get cloer to logrthmc prl whoe lope of the logrthmc curvture grph n () degree () degree 6 (c) degree Fg 7 Typcl cl A Bézer curve nd ther logrthmc curvture grph Typcl cl A Bézer curve re nvrnt under mlrty trnformton To clrfy f curve egment drwn dependng on the poton of the three pont for drwng typcl cl A Bézer curve, we how the drwle regon Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

9 9 n Fg 8 We plce the frt pont vector nd the thrd pont vector t [ ] T nd [ ] T repectvely, nd move wthn the rectngle If typcl cl A Bézer curve drwle (thu, the mtrx tfe the cl A condton nd the curvture monotonclly vryng), we drw the pxel correpondng to wth whte If not drwle, we drw the pxel wth lck From Fg 8, we cn ee tht the degree of typcl cl A Bézer curve elevted, the drwle regon get lrger Fg 9 how vrou kn of cuc cl A Bézer curve, ther curvture plot nd logrthmc curvture grph Fg 9() typcl cl A Bézer curve nce δ δ δ δ Fg 9 ()-() how vrou generl cl A Bézer curve whoe δ re except for the one ndcted Fg how exmple of generl cl A Bézer curve of degree 7 nd Generl cl A Bézer curve of degree hgher thn cn lo e generted n mlr mnner () degree () degree 4 (c) degree 5 (d) degree (e) degree Fg 8 Drwle regon of typcl cl A Bézer curve () δ δ δ δ () δ (c) δ 5 (d) δ (e) δ 7 (f) δ 55 (g) δ 7 (h) δ 65 () δ Fg9 Generl cl A Bézer curve of degree nd ther curvture plot nd logrthmc curvture grph Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy

10 () degree 7 δ δ δ δ () degree 7 δ δ δ Fg Generl cl A Bezer curve of degree 7 nd 9 CONCLUSIONS We hve preented method for nterctvely controllng plnr cl A Bézer curve We frt preented method for nterctvely controllng typcl cl A Bézer curve We howed tht the degree of typcl cl A Bézer curve elevted, the curve converge to logrthmc prl Then we preented method for controllng generl cl A Bézer curve y perturng the coeffcent of cl A mtrx uch tht endpont contrnt re tfed From the vewpont of log-ethetc curve, nterctve control of generl cl A Bézer curve egment men trtng from n pproxmton to log-ethetc curve egment whoe α (logrthmc prl) nd then looenng the lnerty of the logrthmc curvture grph Future reerch nclude nvetgtng f the oluton of the optmzton proce of typcl nd generl cl A Bezer curve unque or not, clrfyng how the hpe of generl cl A Bézer curve wll chnge when δ chnged, nd the extenon to pce curve A ponted out y Co pper[], more nvetgton necery for the cl A condton of generl mtrce Extendng the de of cl A Bezer curve to rtonl Bezer curve, B-plne curve, or NURBS curve lo n nteretng prolem For frng plne curve, new frng lgorthm tht ue the lnerty of the logrthmc curvture grph n optmzton contrnt cn e contructed REFERENCES [] Co, J; Wng G: A note on Cl A Bézer curve, Computer Aded Geometrc Degn, 7, do:6/jcgd7 [] Detz, A D; Pper B: Interpolton wth cuc prl, Computer Aded Geometrc Degn, (), 4, 65-8 [] Frn, G: Curve nd Surfce for CAGD, Acdemc Pre, [4] Frn, G: Cl A Bézer Curve, Computer-Aded Geometrc Degn, (7), 6, [5] Frouk, RT; Skkl T: Pythgoren Hodogrph, IBM Journl of Reerch nd Development, 4, 76-75, 99 [6] Frey, W H; Feld, D A: Degnng Bézer conc egment wth monotone curvture, Computer Aded Geometrc Degn, 7(6),, [7] Grey, A; Slmon, S; Aen, E: Modern Dfferentl Geometry of Curve nd Surfce Wth Mthemtc, CRC Pre, 6 [8] Hrd, T; Yohmoto, F; Morym, M: An ethetc curve n the feld of ndutrl degn In: Proceedng of IEEE Sympoum on Vul Lnguge, IEEE Computer Socety Pre, NewYork, 999, 8 47 [9] Hgh, M, Kneko, K, Hok, M: Generton of hgh qulty curve nd urfce wth moothly vryng curvture, Eurogrphc, 79-9, 998 [] Mur, KT: A generl equton of ethetc curve nd t elf-ffnty Computer-Aded Degn nd Applcton, (-4), 6, [] Mneur, Y: A hpe controlled fttng method for Bézer curve, Computer Aded Geometrc Degn, 5(9), , 998 [] Spd, N S; Frey, W H: Controllng the curvture of qudrtc Bézer curve, Computer Aded Geometrc Degn, 9 (), 85-9, 99 [] Wng, Y; Zho, B; Zhng, L; Xu, J; Wng, K; Wng, S: Degnng fr curve ung monotone curvture pece, Computer Aded Geometrc Degn, (5), pp55-57, 4 [4] Yohd, N; Sto, T: Interctve Aethetc Curve Segment, The Vul Computer (Proc of Pcfc Grphc), (9-), 6, [5] Yohd, N; Sto, T: Qu-Aethetc Curve n Rtonl Cuc Bézer Form, Computer Aded Degn nd Applcton, 4(-4), 7, [6] Yohmoto, F; Hrd, T: Anly of the chrctertc of curve n nturl nd fctory product In: Proc of the nd IASTED Interntonl Conference on Vulzton, Imgng nd Imge Proceng,, 76 8 Computer-Aded Degn & Applcton, 5(-4), 8, xxx-yyy 5 (c) degree δ δ δ δ (d) degree δ δ δ 5