Paths of C-Bézier and C-B-spline curves

PathsofC-B¶ezierandC-B-splinecurvesMikl¶osHo®mann1,YajuanLi2,GuozhaoWang21InstituteofMathematicsandComputerScience,K¶arolyEszterh¶azyUniversity,H-3300Eger,Hungary,email:ho¯@ektf.hu2DepartmentofMathematics,ZhejiangUniversity,Hangzhou,ChinaAbstractC-B¶ezierandC-B-splinecurves,asthetrigonometricextensionsofcubicuniformsplinecurvesarewell-knowningeometricmodeling.Thesecurvesdependonashapeparameter®inawaythat®!0yieldsthecubicpolynomialcurves.Thegeometrice®ectofthealterationofthisparameterisdiscussedinthispaperbythehelpofrelativeparametrizationandlinearapproximation.Keywords:C-B¶eziercurves,C-B-splinecurves,paths1IntroductionInthelastdecadeseveralnewtypesofsplinecurvesandsurfaceshavebeenintroducedtoCAGD.C-curvesareextensionsofthewidelyusedcubicsplinecurvesandareintroducedin[1]byusingthebasissint;cost;t;1.InthecaseofC-B-splinesthisextensioncoincideswiththehelixsplinesde¯nedby[9].Thesetoolsprovideexactrepresentationsofseveralimportantcurvesandsurfacessuchasthecircleandthecylinder[1],theellipse[3],thesphere[6],thecycloidandthehelix[5].FurtherpropertiesofC-curveshavebeenstudiedin[4]and[11].C-curvesareallde¯nedontheintervalt2[0;®],where®2(0;¼]isagivenrealvalue.Since®appearsinallthebasisfunctions,itheavilya®ectstheshapeofthecurve.Whileitisalreadyprovedin[1],thatthelimitingcase®!0isacubicpolynomialcurve,thee®ectsofthemodi¯cationof®havenotbeendescribedyet.Theaimofthispaperistogiveageometricinterpretationofthechangeof®forC-B¶ezierandC-B-splinecurves.Modifyingoneormoredataofagivensplinecurve,thepointsofthecurvewillmoveoncertaincurvescalledpaths.ForexamplemovingoneofthecontrolPreprintsubmittedtoElsevierPreprint28October2005pointsofaB¶ezierorB-splinecurvethesepathswillbeparallellinesegments,whilechangingaweightofaNURBScurvepointsofthiscurvewillmovetowardsthespeci¯edcontrolpointalonglinesegments[10].Alterationofaknotvalueofanon-uniformB-splinecurveyieldswell-de¯nedrationalcurvesaspaths[8].Changingoftheparameter®ofaC-curvethepointsofthecurvewillobviouslychangetheirpositionsaswell.InthispaperthesepathsofC-B¶ezierandC-B-splinecurveswillbediscussed.Thesepathscancloselybeapproximatedbylinesandhavesomenicegeometricpropertieswhichmayyieldtoabetterunderstandingoftheroleof®intermsoftheshapeofthesecurves.2PathsofC-B¶eziercurvesandtheirextensionsConsidertheC-B¶eziercurve(c.f.[1]):b(t;®)=3Xi=0Zi(t;®)pi;t2[0;®];®2(0;¼]wherethebasisfunctionsarede¯nedas:M=8:1if®=¼;sin(®)®¡2®¡sin(®)1¡cos(®)otherwiseZ0(t;®)=(®¡t)¡sin(®¡t)®¡sin(®)Z1(t;®)=MÃ1¡cos(®¡t)1¡cos(®)¡(®¡t)¡sin(®¡t)®¡sin(®)!(1)Z2(t;®)=MÃ1¡cos(t)1¡cos(®)¡t¡sin(t)®¡sin(®)!Z3(t;®)=t¡sin(t)®¡sin(®):Wewouldliketodescribethemovementofasinglepointofthecurveastheparameter®changes.AlteringthisparameterwereceiveafamilyofC-B¶eziercurveswithfamilyparameter®.Duetothechangingdomainofde¯nitionthereisnotmuchsensetoexamineapointofthesecurveswith¯xedparame-tert.Insteadweconsiderthepointateachcurveassociatedtotheparameter(®=ratio),whereratio2[1;1)isa¯xedvalue.Thisparameterchangesfromcurvetocurvebutifthedomainofde¯nition[0;®]wouldbenormalizedto[0;1]foreach®,thenthespeci¯edparameter(®=ratio)wouldhavebeentrans-formedtotheconstantvalue(1=ratio).Thiswaywecande¯netherelative2®-pathsofthefamilyofC-B¶eziercurves:s(®;ratio)=3Xi=0Zi(®=ratio)pi;®2(0;¼];ratio2[1;1)where®istherunningparameteralongthepath,whileratioistheparameterofthepathamongthefamilyofpaths(seeFig.1).Fig.1.TwoC-B¶eziercurvesde¯nedbythesamecontrolpolygonandtheirrelative®-pathsNote,thatthebasisfunctionsoftheoriginalC-B¶eziercurvearesymmetricintfortheparametert=®=2,thustherelative®-pathsalsohaveasymmetricpropertyinratiofortheparameterratio=2.Therelative®-pathassociatedtoratio=2canbedescribedbythefunctionsZ0(®;2)=Z3(®;2)=(®=2)¡sin(®=2)®¡sin(®)Z1(®;2)=Z2(®;2)=MÃ1¡cos(®=2)1¡cos(®)¡(®=2)¡sin(®=2)®¡sin(®)!whichobviouslyyieldsthatthispathisapartofthelineconnectedthemid-pointsofp0p3andp1p2.Pathsassociatedto®6=2arenotlinesasonecaneasilyobservebythemathematicalextensionofthepaths(seeFig2.).ThisFig.2.Extensionofthepathsfor®¸¼3extensionisde¯nedbythepointss(®;ratio)=3Xi=0Zi(®=ratio)pi;ratio2[1;1)for®¸¼.WehavetoemphasizethatthesepointsarenotpointsofanyC-B¶eziercurvesandthesubstitutionofthesevaluesof®ismerelyamath-ematicalextension.SimilarextensionhavebeenusedforpathsofB-splinecurvesin[7].3ApproximatelinesofthepathsThepaths,aswehaveseenarenotlines,butintheoriginalinterval®2(0;¼]theycancloselybeapproximatedbylines.Theapproximatelineofthepaths(®;ratio)canbede¯nedbythejointsegmentofthepoints(¼;ratio)ands(0;ratio)(moreprecisely,since®cannotbeequalto0,weconsiderthepointobtainedby®!0inthislattercase).If®=¼andt=¼=ratio,thenM=1Z0(¼=ratio;¼)=(¼¡¼=ratio)¡sin(¼=ratio)¼Z1(¼=ratio;¼)=1+cos(¼=ratio)2¡(¼¡¼=ratio)¡sin(¼=ratio)¼(2)Z2(¼=ratio;¼)=1¡cos(¼=ratio)2¡¼=ratio¡sin(¼=ratio)¼Z3(¼=ratio;¼)=¼=ratio¡sin(¼=ratio)¼:If®!0,thenfromequations(1)weobtainthefollowinglimits(see[1]):Z0lim(ratio)=(ratio¡1)3ratio3Z1lim(ratio)=3(ratio¡1)2ratio3(3)Z2lim(ratio)=3ratio¡1ratio3Z3lim(ratio)=1ratio3:4Finallyjoiningthepoint3Pi=0Zi(¼=ratio;¼)piandthepoint3Pi=0Zilim(ratio)piweobtainafamilyoflinesw