ONAMETHODOFTHERAPID APPROXIMATION OF A CUBIC BÉZIER CURVE BY QUADRATIC BÉZIER CURVES

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1 ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2014 Seria: MATEMATYKA STOSOWANA z. 4 Nr kol Barbara BIŁY Institute of Mathematics Silesian University of Technology ONAMETHODOFTHERAPID APPROXIMATION OF A CUBIC BÉZIER CURVE BY QUADRATIC BÉZIER CURVES Summary. In this paper the approximation of a cubic Bèzier curve by quadratic Bèzier curves is presented. For some method of the choose of quadratic Bèzier curves, the error of approximation, measured with the Frèchet distance is especially simple to calculate. Moreover, this error is easy topredictalsoforpartsofagivencubicbèziercurve.boththesefeatures give us the opportunity, with some caution, of rapid approximation, where required precision is granted. O METODZIE SZYBKIEJ APROKSYMACJI KRZYWEJ BEZIERA TRZECIEGO STOPNIA KRZYWYMI BEZIERA DRUGIEGO STOPNIA Streszczenie. W artykule zaprezentowano aproksymację krzywej Bèziera trzeciego stopnia za pomocą krzywych Bèziera drugiego stopnia. Dla pewnej metody wyboru krzywych Bèziera drugiego stopnia błąd aproksymacji, mierzony odległością Frècheta, jest szczególnie łatwy do policzenia. Co więcej, błąd ten łatwo policzyć także dla fragmentów wyjściowej krzywej. Obie te właściwości, z pewnymi zastrzeżeniami, umożliwiają szybką aproksymację z wymaganą dokładnością Mathematics Subject Classification: 68U05. Keywords: approximation, cubic Bézier curve, quadratic Bézier curve. Corresponding author: B. Biły(Barbara.Bily@polsl.pl). Received:

2 58 B. Biły 1. Preliminaries 1.1. The definitions of quadratic and cubic Bèzier curves WewillcallthequadraticBèziercurve[2]acurvedefinedontheplane0XY by parametric equations of the t variable [ ] [ ] 1 x(t) c 0x c 1x c 2x B 2 = = C 2 T= t y(t) c 0y c 1y c 2y t 2 fort 0,1.SimilarlyacubicBèziercurve[2]isdefinedas [ ] [ ] x(t) c 0x c 1x c 2x c 3x B 3 = = C 3 T= y(t) c 0y c 1y c 2y c 3y wherec 0x,c 1x,c 2x,c 3x,c 0y,c 1y,c 2y,c 3y R. 1 t t 2 t Equivalent form of Bèzier curves Anequivalentformofdefinition,abitmoreintuitive,usespointsonaplane. ThequadraticBèziercurveisdefinedby3points:P 0,P 1 andp 2.ThetransformationmatrixA 2 guaranteethecoordinatesofthesepointsandthecoefficient matrixc 2 aremutuallyinterchangeable [ c 0x c 1x c 2x c 0y c 1y c 2y C 2 = P 2 A 2, ] [ x P0 x P1 x P2 = y P0 y P1 y P2 ] ThecubicBèziercurveisdefinedby4points:P 0,P 1,P 2 andp 3.ThetransformationmatrixA 3 guaranteethecoordinatesofthesepointsandthecoefficient matrixc 3 aremutuallyinterchangeable C 3 = P 3 A 3. TheformofmatrixA 3 willbepresentedlater,seetheequation(3). BothmatricesA 2 anda 3 arenonsingular,thereforetheyareinvertible.thus thecoordinatesofthepointsp 2 canbecomputedfromthecoefficientmatrixc 2, andsimilarlyp 3 fromc 3.

3 OnamethodoftherapidapproximationofacubicBéziercurve AproximationB 3 curvebyb 2 curve 2.1. Method of approximation TheapproximationB 3 curvebyb 2 curverelyonindicating3pointsq 0, Q 1 andq 2 whichspecifythesearchedcurve.let sdefinethemethodhhof approximation: Definition1.GivenisacubicBéziercurve,specifiedby4points:P 0,P 1,P 2 and P 3.TheapproximatingquadraticBéziercurve,specifiedby3points:Q 0,Q 1 and Q 2.ThemethodHHdeterminesthecoordinatesofthesepointsasfollows Q 0 = P 0, Q 1 = 3(P 1+P 2 ) (P 0 +P 3 ), 4 Q 2 = P 3. TheillustrationofthismethodisshownontheFigure1.Inmatrixnotation, theabovemethodisasfollows Q= [ x Q0 x Q1 x Q2 y Q0 y Q1 y Q2 = ] [ = PK H = x P0 x P1 x P2 x P3 y P0 y P1 y P2 y P3 ] ( ). (1) Ascanbeseenfromtheaboveequation,themethodHHhasthefollowing characteristics: bothcurvesbeginsatthesamepoint, bothcurvesendsatthesamepoint. Feature worth mentioning is the third obligatory point common to both curves, parametersetsittot= 1 2. ToshowthepositionofthepointQ 1,weneedtwoauxiliarypointsH 1 ih 2. Let s define them as follows:

4 60 B. Biły H 1 P 1 Q 1 H 2 P 2 P 0 = Q 0 P 3 = Q 2 Fig. 1. Method HH of approximation Rys. 1. Metoda HH aproksymacji Definition2.GivenisacubicBéziercurve,specifiedby4points:P 0,P 1,P 2 ip 3. TheauxiliarypointsH 1 ih 2 canbesetasfollows H 1 = 3P 1 P 0 2 H 2 = 3P 2 P 3 2,. Withthesepoints,thelocationQ 1 iseasytodeterminegeometrically(figure1): 2.2. Approximation error Q 1 = H 1+H 2. 2 TheHausdorffdistanced H [3]canbeusedasanaturalcandidatetoevaluate the conformity between the approximated and approximating curves. Unfortunately, the analytical calculation in the general case does not seem to be possible, moreover numerical methods are time consuming and error prone. For these reasons,thefréchetdistanced F [3]wereusedtoevaluatetheerror.Althoughitis less precise than the Hausdorff distance, because of the following estimate d H d F. But with usage of Fréchet distance we can solve the problem analytically. To presentit,let sintroducethevector h(figure2): h= H 1 H 2.

5 OnamethodoftherapidapproximationofacubicBéziercurve H 1 P 3 P 1 h P 2 P 0 H 2 Fig. 2.Determinationof hvector Rys.2.Określeniewektora h With this help, the error of this method is determined in the following theorem: Theorem 3. The cubic Bézier curve is approximated by quadratic Bèzier curve determinedbyhhmethod.ifthehausdorffdistanced H isanerrorofsuchapproximation,itsupperlimitisequaltothefréchetdistanced F : d H d F = h 6 3. Theproofofthistheoremisgivenin[1]. 3.ThesubdivisionofBèziercurveB 3 inton smaller parts Theorem4.EachsectionofcubicBéziercurveisalsoacubicBéziercurve. Theproofofthistheoremisgiven[1]. Definition5.EverysectionofcubicBéziercurveisamemberofK-familyoriginated by this curve.

6 62 B. Biły Theorem6.LetthecubicBéziercurveB 3 haveavector h.let screateakfamilybydividingb 3 tonequalparts.anysuchcreatedpartisalsoacubicbézier curveb 3,i andhasitsownvector h i.withinsuchk-family,thesevectorshave the relationship hi = h N3, i=1,2,...,n. (2) An equal division should be understood as follows: when converting Bézier curvetoanequivalentalgebraicform,thelengthoftheparameter tisequalfor allnparts t=t i+1 t i = 1 N, wheret i indicatesadividingpoint.thisisaspecialcaseofamoregeneraltheorem on invariant of K-family presented and proven in[1]. The above theorem is very useful in case of exceeding the acceptable error ofapproximation.aftersplittingtheinitialcurveb 3 intonparts,theorem(6) guaranteestoreducetheestimationoferroroftheextentn Mathematical transformations 4.1. The coefficients of the polynomial Atthebeginning,weset4points:P 0,P 1,P 2 andp 3.Wetransformtheminto asetofcoefficientsc,designedtopresentthebéziercurveinthealgebraicform { x(t) = c 0x +c 1x t+c 2x t 2 +c 3x t 3, y(t) = c 0y +c 1y t+c 2y t 2 +c 3y t 3, Todothis,atransformationmatrixAisneeded C 3 = [ c 0x c 1x c 2x c 3x c 0y c 1y c 2y c 3y = P 4 A 3 = [ ] = x P0 x P1 x P2 x P3 y P0 y P1 y P2 y P3 ] (3)

7 OnamethodoftherapidapproximationofacubicBéziercurve Splitting of a Bézier curve ThenextstepistosplitthecubicBéziercurvetoNparts D i = C 3 K i, i=0,1,...,(n 1), (4) wherek i dependsonτ=τ(i,n)andk=k(n): τ k 0 0 K i = K(τ,k)= τ 2 2τk k 2 0, τ 3 3τ 2 k 3τk 2 k 3 where:n numberofparts,k= 1 N,τ=ik,i=0,1,...,(N 1) Approximation of the parts of Béziera curve A transition to the points of quadratic Bézier curve, which approximate these partsbythemethodhh,isobtainedusingthetransitionmatrix K 2 : [ ] x Q0 x Q1 x Q2 Q i = = D i K 2 = y Q0 y Q1 y Q2 [ ] d 0x d 1x d 2x d (1 3x ) =. d 0y d 1y d 2y d 3y Because of the aforementioned reasons, calculating the coordinates of the point Q 2 isunnecessary,thereforeashortenedversionofthetransitionmatrix K 2 isused [ ] x Q0 x Q1 Q i = = D i K 2 = y Q0 y Q1 [ ] 4 4 d 0x d 1x d 2x d (1 3x 0 2 ) =. (5) d 0y d 1y d 2y d 3y Consolidation All the above transformations(3,4,5), covered collectively form a series of calculations Q i = D i K 2 = C 3 K i K 2 = P 4 A 3 K i K 2. (6)

8 64 B. Biły SinceA,K i and K 2 arethesparsematrices,usingtheaboveformulawould be waste of computing time. After eliminating zero elements, the condensed form of the above formula is expressed as follows k= 1 N, fori =N 1,...,1,0 τ = i N, ξ =1 τ, Q 0,i = P 0 ξ 3 Q 1,i =Q 0,i + k 4 [ +3P 1 τξ 2 +3P 2 τ 2 ξ +P 3 τ 3, P 0 ( τ 6τ 2 + k 2 ) +P 1 ( 6 24τ + 9τ 2 3k 2 ) +P 2 ( 12τ 18τ 2 + 3k 2 ) +P 3 ( 6τ 2 k 2 ) In addition, the coordinates of the third point are obtained from the known coordinates of the first point P 3 ifi=n 1, Q 2,i = (8) Q 0,i+1 ifi<n 1. ]. (7) 5. Example 5.1. Drawing constraints Theauthor saimwastopresentanexamplethatrequiresdivisioninton=3 parts. Unfortunately, such an illustration would be unreadable. This is due to the following calculations: SupposethatallthepointsofcubicBéziercurvehavetofitinacertaindrawing space,limitedbyacircleofradiusr.thenthelargestfréchetdistancewouldbe d F = 3R 6 3 0,29R. WhendividedintoN=3parts,approximationerrorwoulddecreaseatleastN 3 times d F(1/3) =d F /N 3 =d F /27 0,01R.

9 OnamethodoftherapidapproximationofacubicBéziercurve Since the Hausdorff distance, which is the real measure of the distance between curves, would be even smaller, therefore approximating curves would overshadow the approximated curve. Thus, we present an example to illustrate a method for the N=2,wheretheapproximationisnotasgood,andthenbrieflyshowanalogous calculationsforthen= Computational limitations Typically, the input data are selected in such a way that the calculations were as simple as possible. Therefore the following points forming the Bézier curve were selected P 0 [0,0],P 1 [0,2],P 2 [4,1],P 3 [3,3]. Unfortunately, in the course of the calculations it is necessary to divide the coordinatesofthesepointsby4,andbyn 3 forn=2andn=3.thepostulateof avoiding fractions and decimal expansions resulted in need to multiply the initial data by the scale which is the product following factors m= =864 which led to very large coordinates of points P 0 [0,0],P 1 [0,1728],P 2 [3456,864],P 3 [2592,2592] Formulation of the task For4pointsintheplane P 0 [0,0],P 1 [0,1728],P 2 [3456,864],P 3 [2592,2592] depicting cubic Bézier curve and for a given accuracy of approximation inthefirstvariant,and in the second variant, determine: d H =86,4 (=m/10) d H =17,28 (=m/50) N count of quadratic Bézier curves that guarantee an approximation of specifiedaccuracyd H ; listk 1,K 2,...,K N ofthesecurves.

10 66 B. Biły 5.4.SolutionforN= Calculation of the number of curves Wecalculatetheauxiliaryvector h(figure2): h =[hx,h y ] =[3(P 1x +P 2x ) (P 0x +P 3x ),3(P 1y +P 2y ) (P 0y +P 3y )]. In the presented task h=[3888, 2592]. ThenwedeterminethelengthofL h : L h = h = ItisusedtoestimatethesizeoftheN,withtheadditionalsimplifyingassumption d H =d F =86,4: L N 3 h d F thus ListofN=2curves N=2. We find searched quadratic Bézier curves by giving the coordinates of three definingpoints:q 0,Q 1 iq 2.Nextcurves,markedwithi,willreferwithpoints markedasq 0,i,Q 1,i andq 2,i. Fori=0,1,...,N 1wespecifyanauxiliaryvariableτ i = i N.Thenthe startingpointsq 0,i ofsearchedcurvesk i wefindasfollows P 0 [ ] Q 0,i = 1 τ i τi 2 τi P P Withintroductionofanadditionalvariablek= 1 N,thesecondpointQ 1,iwhich specifiessearchcurvesk i iscalculatedasfollows P 0 Q 1,i =Q 0,i + k ] [1 τ i τi 2 k P P P 3 P 3

11 OnamethodoftherapidapproximationofacubicBéziercurve Sincetheendsofcurvesarealsothebeginningsofthenextones,sothereisnoneed ofspecificcomputation.toavoidunnecessarycalculationsq 2 canbedetermined as follows { P 3 fori=n 1, Q 2,i = Q 0,i+1 fori<n 1. Aforementioned observations force the order of evaluation by decreasing indexi.therefore,thecalculationstartsfromi=n 1=1: k= 1 N τ 2 = i N = 1 2, = 1 2, Q 0,1 =[ 1620, 1296], Q 1,1 =[ 2997, 1458], Q 2,1 =P 3 =[ 2592, 2592]. Forthesecondandthelastcurvei=0: τ 0 =0, Q 0,0 =[ 0, 0], Q 1,0 =[ 243, 1134], Q 2,0 =Q 0,1 =[ 1620, 1296]. The Figure 3 shows the result of approximation. P 1 P 3 = Q 2,1 Q 1,1 Q 2,0 =Q 0,1 Q 1,0 h2 h2 P 2 P 0 = Q 0,0 Fig. 3.ApproximationforN=2 Rys.3.AproksymacjadlaN=2

12 68 B. Biły 5.5.SolutionforN= Calculation of the number of curves Knowingvector h(figure4): h=[3888, 2592]. andhislengthl h : L h = h weestimaten,withtheadditionalsimplifyingassumptiond H =d F =17,28: L N 3 h d F hencewegetn: N=3. P 3 = Q 2,2 Q 2,2 Q 1,1 Q 2,1 = Q 0,2 h3 Q 1,0 h3 Q 2,0 =Q 0,1 h3 P 0 = Q 0,0 Fig. 4.ApproximationforN=3 Rys.4.AproksymacjadlaN=3

13 OnamethodoftherapidapproximationofacubicBéziercurve ListofN=3curves Analogouscalculationsstartfromi=N 1=2: k= 1 N τ 2 = i N = 1 3, = 2 3, Forthesecondcurvei=1,thus Q 0,2 =[ 2304, 1536], Q 1,2 =[ 2952, 1776], Q 2,2 =P 3 =[ 2592, 2592]. τ 1 = 1 3, Q 0,1 =[ 864, 1056], Q 1,1 =[ 1656, 1296], Q 2,1 =Q 0,2 =[ 2304, 1536]. Fotthethirdandthelastcurvei=0: τ 0 =0, Q 0,0 =[ 0, 0], Q 1,0 =[ 72, 816], Q 2,0 =Q 0,1 =[ 864, 1056]. TheresultsareshownintheFigure4.Asexpected,theapproximationisso good that it is difficult to distinguish approximating curves from the approximated one Cusp case Letusconsideracasewherethereisacuspcurve,whereinthecurvatureis k= (Figure5,caseA). ItisknownthatnondegeneratequadraticBéziercurvedonothavesuch a cusp. To improve the assessment of the situation, approximating curves have been slightly shifted vertically. As shown, case B, where division point falls on acusp,isdifficulttoaccept.ifithappens,itisrequiredtoincreasethenumberof divisions by 1, which ensures that the jump over an obstacle. Emerging situation issimilartothevariantc.iftheresultsarenotsatisfactoryagain,onecanreduce the acceptable error, by increasing the number of approximating curves. Of course, itisnecessarytocheckwhethertheincreasedndoesnotrepeatthesituationb.

14 70 B. Biły A B, N = 2 C, N = 3 Fig. 5.Cuspcase Rys. 5. Przypadek ze szpicem The same objections may arise during the approximation of Bèzier curves whichcontainsaloop.here,too,itisrecommendedtodecreasetheerrorofapproximation. Fortunately, in practice such cases does not occur, so their in-depth analysis is not a pressing problem. 6. Concluding remarks The presented method can be used in some environments to create graphics. IftheenvironmentdoesnothavetheabilitytocreatecubicBéziercurves,andcan create quadratic Bézier curves, then the method can be useful. Such environments are, for example, Adobe Flash, which uses SWF file format[4], currently very popular in the Internet. References 1. Biły B.: Béziera Curves. Selected Elements of the Theory with Examples. WPKJS, Gliwice 2012(in Polish).

15 OnamethodoftherapidapproximationofacubicBéziercurve Foley J.D., van Dam A., Feiner S.K., Hughes J.F., Phillips R.L.: Introduction to Computer Graphics. WNT, Warszawa 1995(in Polish). 3. Kuratowski K.: Introduction to Set Theory and Topology. PWN, Warszawa 1980 (in Polish) #main Quadratic Beziers and Cubic Beziers.

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