Themethodofmovingcurvesandmovingsurfacesisanew,eectivetoolfor Abstract

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1 OnaRelationshipbetweentheMovingLineand MovingConicCoecientMatrices DepartmentofComputerScience Houston,Texas77005 RiceUniversity MingZhang DepartmentofInformationSystemsandComputerScience NationalUniversityofSingapore KentRidge,Singapore Eng-WeeChionh DepartmentofComputerScience Houston,Texas77005 RonaldN.Goldman RiceUniversity tweenthemovinglinecoecientmatrixandthemovingconiccoecientmatrixfor implicitizingrationalcurvesandsurfaces.hereweinvestigatearelationshipbe- Themethodofmovingcurvesandmovingsurfacesisanew,eectivetoolfor Abstract Keywords:Implicitization;MovingLine;MovingConic rationalcurves.basedonthisrelationship,wepresentanewproofthatthemethod ofmovingconicsalwaysproducestheimplicitequationofarationalcurvewhen therearenolowdegreemovinglinesthatfollowthecurve.

2 1Introduction Foreachrationalcurve wherex(t);y(t);w(t)aredegreenpolynomialsintandgcd(x(t);y(t);w(t))=1,there existsauniqueirreducibledegreenpolynomialf(x;y)suchthatf(x;y)=0represents thesamecurveas(1).theequationsx=x(t)=w(t),y=y(t)=w(t)arecalledthe parametricformofthecurve,whereastheequationf(x;y)=0iscalledtheimplicit formofthesamecurve.surfacestoohaveparametricandimplicitforms.parametric w(t); w(t); representationsareconvenientforrenderingcurvesandsurfaces,whiletheimplicitforms areusefulforcheckingwhetherornotapointliesonacurveorsurface.implicitization istheprocessofndingtheimplicitrepresentationsforcurvesorsurfacesfromtheir parametricrepresentations. curvesandsurfaceswithbasepoints(sederbergetal1994;sederbergetal1997). ofmovingalgebraiccurvesandsurfacestosolvetheimplicitizationproblemforrational pointof(1)isaparametert0suchthatx(t0)=y(t0)=w(t0)=0.atabasepointthe rationalexpressionsforxandybothhavetheform0.sederbergintroducedthemethod identicallyinthepresenceofbasepoints(chionh1990;manocha&canny1992).abase surfaces(goldmanetal1984;demontaudouin&tiller1984).butresultantsvanish Resultantscanbeappliedtosolvetheimplicitizationproblemforrationalcurvesand Amovinglineofdegreed parametert.similarly,amovingconicofdegreed isaoneparameterfamilyofimplicitlydenedlines,withonelinecorrespondingtoeach Amongthealgebraiccurves,movinglinesandmovingconicsarethemostimportant. dxi=0(aix2+biy2+cixy+dixw+eiyw+fiw2)ti=0 dxi=0(aix+biy+ciw)ti=0 (2) isaoneparameterfamilyofimplicitlydenedconics.amovingline(2)oramoving conic(3)issaidtofollowarationalcurve(1)if dxi=0(aix(t)+biy(t)+ciw(t))ti0; (3) ordxi=0(aix2(t)+biy2(t)+cix(t)y(t)+dix(t)w(t)+eiy(t)w(t)+fiw2(t))ti0:(5) (4) zero,wegeneratealinearsystemwithunknownsfai;bi;ci;dig(orfai;;fig).any BysettingthecoecientsofallmonomialstiinEquation(4)(orEquation(5))to 2

3 (1).Themethodofmovinglines(movingconics)constructstheimplicitequationofa rationalcurvebytakingthedeterminantofthecoecientmatrixofasetofindependent movinglines(movingconics)thatfollowthecurve.hereindependencemeansnotjust solutionofthislinearsystemisamovingline(orconic)thatfollowstherationalcurve thelinearindependenceofthesolutionsofthelinearsystemgeneratedfromequation example,kmovinglines (4)(orEquation(5)),butratherindependenceofthemovinglines(orconics).Thus,for aresaidtobeindependentifthematrix lk(ak;0x+bk;0y+ck;0w)++(ak;dx+bk;dy+ck;dw)td=0 l1(a1;0x+b1;0y+c1;0w)++(a1;dx+b1;dy+c1;dw)td=0;. 264A1;0x+B1;0y+C1;0wAk;0x+Bk;0y+Ck;0w studied(sederbergetal1997),andcleverapplicationsofmovinglinesarepresentedin equationofarationalcurve(sederbergetal1997).movinglineshavebeenthoroughly isofrankk. Itisknownthatthemethodofmovinglinesalwayssuccessfullyproducestheimplicit A1;dx+B1;dy+C1;dwAk;dx+Bk;dy+Ck;dw (Coxetal1998;Chionhetal1998).However,themovinglinemethodrequirescomputing alargedeterminanttogeneratetheimplicitrepresentation.incontrasttothemethod anadvantageoverthemovinglinesmethod,sincethemethodofmovingconicscomputes ofmovinglines,themethodofmovingconicsdoesnotalwayssuccessfullyyieldthe implicitequationofarationalcurve.rather,themovingconicsmethodproducesthe theimplicitequationofarationalcurvebytakingadeterminantofmuchsmallersize followingthecurve(sederbergetal1997).nevertheless,themovingconicsmethodhas thanthedeterminantgeneratedbythemethodofmovinglines. implicitequationofarationalcurveifandonlyiftherearenolowdegreemovinglines ofthemethodofmovingconicsforrationalcurvesisessentialtoextendingboththe movingsurfaces,wherebasepointsplayamorefundamentalrole.aclearunderstanding methodandtheproofstorationalsurfaces(c.f.section4). conics.forarationalcurvex=x(t) Ourultimategoalistogeneralizethemethodofmovingcurvestothemethodof Thispaperpresentsanewperspectiveonthemethodsofmovinglinesandmoving matrix.inparticularweshallshowthatformovinglinesandmovingconicsofdegree aremovinglines(movingconics)thatfollowtherationalcurve.wecallthecoecient matrixofthislinearsystemthemovinglinematrix(movingconicmatrix).thegoalof equatingthecoecientsofallmonomialsinttozero.thesolutionsofthislinearsystem generatedbysubstitutingx(t);y(t);w(t)intothemovingline(4)(movingconic(5))and thispaperistoderivearelationshipbetweenthemovinglinematrixandthemovingconic w(t),y=y(t) w(t)ofdegree2m,considerthelinearsystem m?1,thedeterminantofthemovinglinematrixactuallyfactorsasub-determinantof 3

4 in(sederbergetal1997),thisnewproofseemstogeneralizenaturallytothemethodof movingquadricsforrationalsurfaces. conicssuccessfullyproducestheimplicitequationoftherationalcurve.unlikethework therearenolowdegreemovinglinesfollowingarationalcurve,themethodofmoving themovingconicmatrix.basedonthisobservation,wepresentanewproofthatwhen matrixnotation,andinsection3weintroducethemovinglineandmovingconicmatrices. Section4establishestherelationshipbetweenthemovinglineandmovingconicmatrices torationalcurvesofodddegrees,andbrieydiscusspossiblegeneralizationstorational movingconicssuccessfullygeneratestheimplicitequationofarationalcurvewhenthere isnolowdegreemovinglinethatfollowsthecurve.insection5,weextendtheseresults forrationalcurvesofevendegrees,andusesthisrelationshiptoprovethatthemethodof Therestofthispaperisorganizedinthefollowingway.InSection2wexsome surfaces. by1;t;;td.thatis, denotethe(d+1)kcoecientmatrixofthepolynomialspi(t)whoserowsareindexed degreedi,andletd=max(d1;d2;;dk).weshallwritehp1(t)p2(t)pk(t)icto 2Notation Forconvenience,weadoptthefollowingnotation:Letpi(t),1ik,bepolynomialsof Forexample,h(1+2t)(1+2t)t3(4+5t+6t2)iC= p1(t);;pk(t)=(1td)hp1(t)p2(t)pk(t)ic: Ifhp1(t)p2(t)pk(t)iCisasquarematrix,wedenoteitsdeterminantby 205 p1(t)pk(t): : InSections3and4,wewillstudyonlyrationalcurveswithevendegreesandnobase 3MovingLineandMovingConicMatrices points.considerthenadegree2mrationalcurveinhomogeneousformx(t):y(t):w(t), 4

5 andgcd(x(t);y(t);w(t))=1.thecartesiancoordinatesofpointsonthecurvearegiven where x(t)=2mxi=0aiti;y(t)=2mxi=0biti;w(t)=2mxi=0citi; X=x(t) w(t); Y=y(t) w(t): (6) generatedbyequatingthecoecientsofallmonomials(int)inequation(4)tozero.we Thecoecientmatrixhx;y;w,,tdx;tdy;tdwiCisofsize(2m+d+1)(3d+3). canwritethissystemas Tondadegreedmovinglinethatfollowscurve(6),weconsiderthelinearsystem Similarly,tonddegreedmovingconicsthatfollowcurve(6),weconsiderthelinear systemgeneratedbyequatingthecoecientsofallmonomials(int)inequation(5)to zero.wecanwritethissystemas hx;y;w;;tdx;tdy;tdwic[a0;b0;c0;;ad;bd;cd]t0: hx2;y2;xy;xw;yw;w2;;tdxw;tdyw;tdw2ic[a0;b0;c0;;dd;ed;fd]t0:(8) (7) Thecoecientmatrixhx2;y2;xy;xw;yw;w2,,tdxw;tdyw;tdw2iCisofsize(4m+d+ 1)(6d+6). degreem?1thatfollowthecurveandthengeneratestheimplicitrepresentationfrom inx;y.themethodofmovingconicsndsasetofmindependentmovingconicsof thedeterminantofthecoecientmatrixofthissetofmovingconics.considerthenm independentmovingconicsofdegreem?1thatfollowthecurve: Itiswellknownthattheimplicitformofcurve(6)isapolynomialofdegree2m whereci;j(x;y;w),0i;jm?1,arequadraticinx;y;w.itisknownthatsuch independentconicsexistandthat pm?1(x;y;w;t)=cm?1;0(x;y;w)+cm?1;1(x;y;w)t++cm?1;m?1(x;y;w)tm?1; p0(x;y;w;t)=c0;0(x;y;w)+c0;1(x;y;w)t++c0;m?1(x;y;w)tm?1; c0;0(x;y;w)cm?1;0(x;y;w). istheimplicitequationofcurve(6),whentherearenomovinglinesofdegreem?1 thatfollowthecurve(sederbergetal1997).wewillthereforeconsiderthecasewhere d=m?1inequations(7)and(8).then[x;y;w,,tdx;tdy;tdwicisasquarematrix c0;m?1(x;y;w)cm?1;m?1(x;y;w)=0... 5

6 oforder3m denotethis3m3mmatrixbyml.thematrixhx2;y2;xy;xw;yw;w2,,tdxw;tdyw;tdw2icisofsize5m6m denotethis5m6mmatrixbymc.our showthatthemethodofmovingconicsworkswhenevertherearenolowdegreemoving linesthatfollowthecurve. goalistondarelationshipbetweenmlandmc,andthentoapplythisrelationshipto MCw=hx2y2xyxwywtm?1x2tm?1y2tm?1xytm?1xwtm?1ywiC5m5m: 5m.Wedenotethis5m5msubmatrixbyMCw.Tosummarize: polynomialstkw2;k=0;;m?1.thisproceduregeneratesasquaresubmatrixoforder ToobtainasquaresubmatrixfromMC,wedeletethecolumnsthatrepresentthe ML=hxywtm?1xtm?1ytm?1wiC3m3m; fewpreliminarylemmas. Lemma1jMLjisirreducibleinthecoecientsofx(t);y(t);w(t)(Sederbergetal1997). 4jMLjFactorsjMCwj WearegoingtoshowthatjMCwj=constantResulant(x;y)jMLj2.Webeginwitha (Macaulay1916). Lemma3IfjMLj=0,thenjMCwj=0. Lemma2TheresultantRx(t);y(t)isirreducibleinthecoecientsofx(t);y(t) arelinearlydependent.therefore,jmcwj=0.2 thepolynomialstkx2,tky2,tkxy,tkxw;tkyw,k=0;;m?1.thusthecolumnsofmcw Proof.IfjMLj=0,thenthecolumnsinMLarelinearlydependent.Thatis,thereexist constantsi;i=1;;3m,suchthat MultiplyingbothsidesofEquation(9)byx(t)ory(t),wegetalinearrelationshipbetween 1x(t)+2y(t)+3w(t)++3m?2tm?1x(t)+3m?1tm?1y(t)+3mtm?1w(t)0:(9) Lemma4IftheresultantRx(t);y(t)=0,thenjMCwj=0. 6

7 commonroott0.therefore, Proof.WhentheresultantRx(t);y(t)=0,thetwopolynomialsx(t);y(t)havea =[x2(t0)y(t0)w(t0)tm?1 [1t0t5m?1 0]MCw Lemma5IfjMCwj=0,theneitherRx(t);y(t)=0orjMLj=0. ThustherowsofMCwarelinearlydependent,sojMCwj=0.2 =[00]15m: 0x2(t0)tm?1 0y(t0)w(t0)]15m Proof.IfjMCwj=0,thenthereexistconstantsi;i=1;;5msuchthat Collectthecoecientsofx2;y2;xy;xw;ywandrewriteEquation(10)as wherepi(t);i=1;;5arepolynomialsintofdegreem?1.equation(11)canalsobe writtenas 1x2(t)+2y2(t)+3x(t)y(t)++5m?1tm?1x(t)w(t)+5my(t)w(t)0:(10) p1(t)x+p3(t)y+p4(t)wx?p2(t)y?p5(t)wy: p1(t)x2+p2(t)y2+p3(t)xy+p4(t)xw+p5(t)yw0; (11) thanorequaltom?1suchthatq(t)x=?p2(t)y?p5(t)w.thatis, wecanassumedegree(x)=2m.therefore,thereexistsapolynomialq(t)ofdegreeless p1(t)x+p3(t)y+p4(t)w,andxmustdivide?p2(t)y?p5(t)w.withoutloseofgenerality, x(t)andy(t)donothaveacommonroot.therefore,fromequation(12),ymustdivide NowweshallprovethatifRx(t);y(t)6=0,thenjMLj=0.IfRx(t);y(t)6=0,then (12) p1(t);p3(t);p4(t),suchthatp1(t)x+p3(t)y+p4(t)w0 Ifp2(t)0andp5(t)0,thenbyEquation(12)thereexistnon-zeropolynomials MLarelinearlydependent,sincethedegreesofq(t);p2(t);p5(t)areallatmostm?1. Whenp2(t)orp5(t)arenotidenticallyzero,relationship(13)assertsthatthecolumnsof q(t)x+p2(t)y+p5(t)w0: (14) (13) jmlj=0.2 againassertsthatthecolumnsofmlarelinearlydependent.therefore,ineithercase, withthedegreesofp1(t);p3(t);p4(t)alllessthanorequaltom?1.thisrelationship Theorem6jMCwj=cRx(t);y(t)jMLj2,wherecissomenon-zeroconstant. Withallthispreparation,wecannownallyproveourmainresult. 7

8 Proof.FromLemma5,weconcludethatjMCwjhasonlytwonon-constantfactors: x(t);y(t);w(t).first,considerjmcwj.sinceeachcolumnofmcwcontainsentrieshomogeneousinthecoecientsofx(t);y(t);w(t),thedeterminantjmcwjisahomogeneous wherecissomenon-zeroconstant. polynomialinthesecoecients.specically,jmcwjishomogeneousofdegree4minthe Letusnowexaminethedegreesofeachofthesedeterminantsinthecoecientsof jmcwj=chr(x(t);y(t)ipjmljq: (15) R(x(t);y(t))andjMLj.Therefore,thereexistpositiveintegersp;qsuchthat coecientsofx(t)andy(t),andhomogeneousofdegree2minthecoecientsofw(t). Ontheotherhand,theresultantRx(t);y(t)ishomogeneousofdegree2minthecoecientsofx(t)andy(t),andjMLjishomogeneousofdegreeminthecoecientsof x(t);y(t);w(t). Comparingthehomogeneousdegreesinthecoecientsofx(t);y(t);w(t)onbothsides ofequation(15),wehavethefollowingequalities: Itiseasytoseethattheonlysolutiontotheseequalitiesisp=1andq=2.Theproof isthereforecomplete.2 2mp+mq=4m; mq=2m:(inthecoecientsofw(t)) (inthecoecientsofx(t)) thecolumnsthatrepresentthepolynomialstkx2ortky2,0km?1,wecanobtain NotethatintheoriginalmovingconicmatrixMC(c.f.Equation(8)),ifwediscard (inthecoecientsofy(t)) similarresults: Corollary7Ifadegree2mrationalcurvex(t):y(t):w(t)doesnothavebasepoints, jmcxj=cry(t);w(t)jmlj2; andtherearenodegreem?1movinglinesthatfollowx(t):y(t):w(t),thenthemethod jmcyj=crw(t);x(t)jmlj2: ofmovingconicsalwayssucceedsinproducingtheimplicitequationforx(t):y(t):w(t). rootamongx(t);y(t);w(t).wecanthentranslatethecurvex(t):y(t):w(t)to x(t0)=0;y(t0)=0,weknowthatw(t0)6=0since,byassumption,thereisnocommon thesuccessofthemovingconicsmethod.infact,iftheresultantrx(t);y(t)=0, thenx(t)andy(t)haveatleastonecommonroot.butforanyparametert0suchthat Proof:FirstweobservethatthevanishingoftheresultantRx(t);y(t)doesnotaect 8

9 resultantrx(t);y(t)isnotzero. theimplicitequationoftheoriginalcurveisequivalenttondingtheimplicitequation ofthetranslatedcurve.thuswecanalwaysassume,withoutloseofgenerality,thatthe implicitequationofthetranslatedcurveisf(x+constant;y)=0.therefore,tond commonroots.iff(x;y)=0istheimplicitequationoftheoriginalcurve,thenthe x(t)+constantw(t):y(t):w(t)sothatx(t)+constantw(t)andy(t)donothave x(t):y(t):w(t),thenjmlj6=0.thereforebytheorem7,jmcwj6=0.writethe linearsystem(8)(whend=m?1)as Second,iftherearenodegreem?1movinglinesthatfollowtherationalcurve MCw2 6 4Am?1 A E0. 0 The5mmmatrix[w2tm?1w2]ContherighthandsideofEquation(16)has Em?1 3 75=?[w2tm?1w2]C264F0 fullrankm,becausethecolumnsofthismatrixarelinearlyindependent.therefore,the Fm?1375:. system(16)hasmlinearlyindependentsolutions.letpi,0im?1,bethesolution ofsystem(16)correspondingtosetting tries.hencethedeterminantjp0pm?1jcontainsthetermw2m.thus,thisdeter- minantdoesnotvanishidentically.sinceeachentryinthisdeterminantisquadraticin Thereforethecoecientmatrix[P0Pm?1]Ccontainsw2onlyinthediagonalen- 0j6=i: Then Pi=tiw2+termswithoutw2; Fj=(1j=i; x;y;w,thetotaldegreeofthisdeterminantisatmost2m.moreover,byconstruction, eachcolumnpi,0im?1,isamovingconicthatfollowsthecurve,soforpoints onthecurve,therowsarelinearlydependent;hencethisdeterminantiszeroforpoints 0im?1: m?1thatfollowsthecurve.2 x(t):y(t):w(t)isrepresentedbyauniqueirreducibledegree2mpolynomialequation. onthecurve.ontheotherhand,theimplicitequationofthedegree2mrationalcurve Therefore,thedeterminantjP0Pm?1jmustbetheimplicitequationoftherational curve,sothemethodofmovingconicssucceedswhenthereisnomovinglineofdegree 9

10 5GeneralizationsandExtensions 5.1RationalCurveswithOddDegrees InSections3and4,wediscussedarelationshipbetweenthemovinglineandmoving propositionshold. conicmatricesforrationalcurvesofevendegrees.forodddegreerationalcurves,similar isofsize(3m+2)(3m+3).togetasquaresubmatrix,discardthelastcolumnfrom ML,andwritetheresultingsubmatrixas Considerarationalcurvex(t):y(t):w(t)ofdegree2m+1.Themovinglinematrix MLw=[xywtmxtmy]C(3m+2)(3m+2): ML=[xywtmxtmytmw]C km,andtmxw;tmyw.theresultingsquaresubmatrixis isofsize(5m+3)(6m+6).deletethecolumnsthatrepresentthepolynomialstkw2;0 Themovingconicmatrix MC=[x2y2xyxwyww2tmx2tmy2tmxytmxwtmywtmw2]C wherecissomenon-zeroconstant.itfollowsfromthisequationbyanargumentanalogous ByananalysissimilartothatofSection4,wehave MCw=[x2y2xyxwywtmx2tmy2tmxy]C(5m+3)(5m+3): tothatintheproofofcorollary7thatwhentherearenobasepointsandwhenthere curvex(t):y(t):w(t),themethodofmovingconicssuccessfullygeneratestheimplicit existsonlyoneindependentmovinglineofdegreemthatfollowsthedegree2m+1 equationfortherationalcurve. jmcwj=cresultantx(t);y(t)jmlwj2; 5.2RationalSurfaces Forrationaltensorproductsurfaces,movinglinesandmovingconicsgeneralizetomovingplanesandmovingquadrics.Empiricalstudiesandnumericalexperimentsshowthat surface.moreover,usingthismethodtheimplicitequationofasurfaceofbidegree(m;n) themethodofmovingquadricsgenerallyproducestheimplicitequationforarational bytheusualresultantmethods(order2mn).furthermore,whenbasepointsarepresent, isrepresentedbyamuchmorecompactdeterminant(ordermn)thantheonegenerated 10

11 letmqbethemovingquadriccoecientmatrix.weconjecturethat veryusefultoknowexactlywhenthemethodofmovingquadricsworks. simpliesinthepresenceofbasepoints(sederberg&chen1995).therefore,itwouldbe standardresultantmethodseitherfail(becomeidenticallyzero)orbecomeverycomplicated,whereasthemethodofmovingquadricsstillgenerallysucceedsandindeedoften wheremqwisthesubmatrixofmqobtainedbydeletingthecolumnsrepresentingthe amongthecolumnsofmlgeneratestworelationsamongthecolumnsofmcw(multiplyingbyx(t)ory(t))[c.f.lemma3];hencejmljisadoublefactorofjmcwj.forthe bivariatecase,eachrelationamongthecolumnsofmpgeneratesthreerelationsamong jmqwj=cresultant(x(s;t);y(s;t);z(s;t))jmpj3; polynomialsthataremultiplesofw2.notethatintheunivariatesetting,eachrelation Inanalogywithrationalcurves,letMPbethemovingplanecoecientmatrix,and thecolumnsofmqw(multiplyingbyx(s;t),y(s;t)orz(s;t)).soweexpectjmpjshould surface. deedholdforrationalsurfaces.wehopetoprovethisassertioninafuturepaper,and beatriplefactorofjmqwj.numericalexperimentsshowthatthisrelationshipdoesin- Acknowledgments toapplythisresulttoshowthatthemethodofmovingquadricsalwayssuccessfullyimplicitizesarationalsurfacewhentherearenolowdegreemovingplanesthatfollowthe providedbybyu.mingzhangandrongoldmanarepartiallysupportedbynsfgrant CCR BrighamYoungUniversity.Hegreatlyappreciatesthehospitalityandfacilitiesgenerously Eng-WeeChionhissupportedbytheNationalUniversityofSingaporeforresearchat References [1]Chionh,E.W.,Zhang,M.,Goldman,R.N.(1998),ImplicitizationMatricesinthe [3]Goldman,R.N.,Sederberg,T.,Anderson,D.(1984),VectorElimination:ATech- [2]Cox,D.,Sederberg,T.W.,Chen,F.(1998),TheMovingLineIdealBasisforPlanar StyleofSylvesterwiththeOrderofBezout,submittedtoComputerAidedDesign. [4]Macaulay,F.S.(1916),TheAlgebraicTheoryofModularSystems,CambridgeUniversityPress. RationalCurves,ComputerAidedGeometricDesign,toappear. niquefortheimplicitization,inversion,andintersectionofplanarparametricratio- nalpolynomialcurves.computeraidedgeometricdesign1,

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