: However, such a definition is unintelligible 1972, Forrest published his famous paper inComputer Aided Design journal,- he pointed out that, Bezier curve can be definedin terms of points with help of BernsteinPolynomials.- Liang Youdong, Chang Gengzhe, Liu Dingyuan- P Bézier was passed away in 1999, CAGDpublished a Special issue for him in 2001
• However, such a definition is unintelligible • 1972, Forrest published his famous paper in Computer Aided Design journal, – he pointed out that, Bézier curve can be defined in terms of points with help of Bernstein Polynomials. – Liang Youdong, Chang Gengzhe, Liu Dingyuan – P Bézier was passed away in 1999, CAGD published a Special issue for him in 2001
CAGD special issueCOMPUTERAIDEDGEOMETRICDESIGNELSEVIERComputerAidedGeometricDesign18 (2001)667-671www.elsevier.com/locate/comaidConversionbetweentriangularandrectangularBezierpatchesShi-MinHuDepartmentofComputerScienceandTechnology,TsinghuaUniversty,Beljing100084,PRChinaReceived September 2000; revised May 2001InmemoryofPBezierAbstractThis paper presents an explicit formula that converts a triangular Bezier patch of degree n to adegeneraterectangularBezier patch of degreen Xn by reparametrizationBased on this formula, wedevelop a method for approximating a degenerate rectangularBezier patch by threenondegenerateBezier patches, more patches can be introduced by subdivision to meet a user-specified errortolerance.@2001ElsevierScienceB.V.All rightsreservedKenwords:Bezier surfaces;Degreeelevation, Subdivision,Conversion
CAGD special issue
Definition: Given control points Po,Pi,... Pn, Bezier curvecan be defined as:ht e[0,1]P(t) = ZP,Bi,n(t),i=0Where Bin(t) is i-th Bernstein polynomial ofdegree nn!n-Bi,n(t) = Cti(1-t)n-i(1-il(n-i)!(i = O,1,...n)
Definition • Given control points P0,P1,.Pn, Bezier curve can be defined as: Where B (t) is i-th Bernstein polynomial of degree n ( ) ( ), [0,1] , 0 P t P B t t i i n n i ( 0,1,. ) (1 ) !( )! ! ( ) (1 ) , i n t t i n i n B t C t t i i n i i n i i n n
OPOP2OP3.Pt=0Dt=0OODegreethreeDegreetwoOPoPot=0OP3DegreefourThreeBeziercurves
Three Bezier curves Degree two Degree three Degree four
Property of Bernstein polynomial·Non-negative (非负)=0t=0,1Bin(t) =>0te(0,1),i= 1,2,..,n- 1;· End point1(i= 0)B(0)0otherswise[1(i=n)B10otherswise
Property of Bernstein polynomial • Non-negative ( ) • End point , 0 0,1 ( ) 0 (0,1), 1,2, , 1; i n t B t t i n otherswise i n B otherswise i B i n i n 0 1 ( ) (1) 0 1 ( 0) (0) ,