贝塞尔插值的制作方法

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1、BéziercurvesWrittenby PaulBourkeOriginal:April1989,Updated:December1996ThefollowingdescribesthemathematicsforthesocalledBéziercurve.ItisattributedandnamedafteraFrenchengineer,PierreBézier,whousedthemforthebodydesignoftheRenaultcarinthe1970's.Theyhavesi

2、nceobtaineddominanceinthetypesettingindustryandinparticularwiththeAdobePostscriptandfontproducts.ConsiderN+1controlpointspk(k=0toN)in3space.TheBézierparametriccurvefunctionisoftheformB(u)isacontinuousfunctionin3spacedefiningthecurvewithNdiscretecontrol

3、pointsPk.u=0atthefirstcontrolpoint(k=0)andu=1atthelastcontrolpoint(k=N).Notes:·Thecurveingeneraldoesnotpassthroughanyofthecontrolpointsexceptthefirstandlast.Fromtheformula B(0)=P0 and B(1)=PN.·Thecurveisalwayscontainedwithintheconvexhullofthecontrolpoi

4、nts,itneveroscillateswildlyawayfromthecontrolpoints.·IfthereisonlyonecontrolpointP0,ie:N=0then B(u)=P0 forallu.·IfthereareonlytwocontrolpointsP0 andP1,ie:N=1thentheformulareducestoalinesegmentbetweenthetwocontrolpoints.·thetermiscalledablendingfunction

5、sinceitblendsthecontrolpointstoformtheBéziercurve.·Theblendingfunctionisalwaysapolynomialonedegreelessthanthenumberofcontrolpoints.Thus3controlpointsresultsinaparabola,4controlpointsacubiccurveetc.·Closedcurvescanbegeneratedbymakingthelastcontrolpointt

6、hesameasthefirstcontrolpoint.Firstordercontinuitycanbeachievedbyensuringthetangentbetweenthefirsttwopointsandthelasttwopointsarethesame.·Addingmultiplecontrolpointsatasinglepositioninspacewilladdmoreweighttothatpoint"pulling"theBéziercurvetowardsit.·As

7、thenumberofcontrolpointsincreasesitisnecessarytohavehigherorderpolynomialsandpossiblyhigherfactorials.ItiscommonthereforetopiecetogethersmallsectionsofBéziercurvestoformalongercurve.Thisalsohelpscontrollocalconditions,normallychangingthepositionofoneco

8、ntrolpointwillaffectthewholecurve.Ofcoursesincethecurvestartsandendsatthefirstandlastcontrolpointitiseasytophysicallymatchthesections.Itisalsopossibletomatchthefirstderivativesincethetangentattheendsisalongthelinebetweenthetwopointsatth