A new type of the generalized Bezier curves

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1、App1．Math．J．ChineseUniv2011，26(1)：47-56AnewtypeofthegeneralizedB6ziercurvesCHENJieWANGGuo—jinAbstract．Inthispaper，weimprovethegeneralizedBernsteinbasisfunctionsintroducedbyHan．eta1．ThenewbasisfunctionsnotonlyinheritthemostpropertiesoftheclassicalBernste

2、inbasisfunctions，butalsoreservetheshapeparametersthataresimilartotheshapeparametersofthegeneralizedBernsteinbasisfunctions．Thedegreeelevationalgorithmandtheconver—sionformulaebetweenthenewbasisfunctionsandtheclassicalBernsteinbasisfunctionsareobtained．A

3、lsothenewQ—B~ziercurveandsurfaceconstructedbythenewbasisfunctionsaregivenandtheirpropertiesarediscussed．§1IntroductionInCAGD／CADsystem，parametricrepresentationofcurvesandsurfacesanditsshapeCOntrolareunderwideconsiderationandresearch．anditiSveryimportant

4、tochoosetheap—propriatebasisfunctions．Basedontheirgoodproperties，theclassicalBernsteinbasisfunctionsplayasignificantroleinCAGD／CADsystem．Inrecentyears，theparametricrepresentationofcurvesandsurfaceswithshapeparametershasreceivedattentioninf3,41．Haneta1．[

5、4]introducedanewtypeofthegeneralizedBernsteinbasisfunctionsassociatedwiththeshapeparametersandB~ziercurvesandsurfaceswhichseemstobeentirelynew，andthesecurvesandsurfaceshavesimplerformandclearergeometricmeaningthanrationalBdziercurvesandsurfaces．Notethat

6、themathematicalprecisenessaboutthestatementinf4]shouldbedoubted．Itiswell-knownthatthenumberofbasisfunctionsisn+1inthespaceofpolynomialsofdegreen，thatis，thedegreeofthepolynomialsislowerthanthedimensionofthec0rrespondingpolynomialspacebyone．Butaccorclingt

7、othedefinitionofthebasisfunctionsthere，thedegreeandthenumberofSO—calledbasisfunctionsarethesame．Infact，thoughallfunctionsformingthisgrouparelinearindependenttoeachother，theydon’tconstructamaximallinearlyindependentset．Soitseemstobeinappropriatetocallthe

8、mbasisfunctions．Received：2009．12—29．MRSubjectClassification：65D18．Keywords：Shapeparameter，shapecontrol，degreeelevation．DigitalObiectIdentifier(DoI1：10．1007／sl1766-01i-2453．8．ThisworkwassupportedbytheNationalNaturalScienceFoundati