Distance for degree raising and reduction of triangular Bezier surfaces

Transcription

1 Joural of Computatioal ad Applied Mathematics 158 (003) Distace for degree raisig ad reductio of triagular Bezier surfaces Abedallah Rababah Departmet of Mathematics ad Statistics, Jorda Uiversity of Sciece ad echology, rbid 110, Jorda Received 1 Jauary 003; received i revised form 18 March 003 Abstract he problem of degree reductio ad degree raisig of triagular Bezier surfaces is cosidered.he L ad l measures of distace combied with the least-squares method are used to get a formula for the Bezier poits.he methods use the matrix represetatios of the degree reductio ad degree raisig. c 003 Elsevier B.V. All rights reserved. MSC: 41A65; 65D17; 65D18; 68U05; 68U07 Keywords: Geeralized Berstei polyomials; riagular Bezier surfaces; Degree reductio; Degree raisig; Computeraided geometric desig 1. troductio the applicatios of CAGD, we may have cotrol poits which do ot t i a rectagular domai, ad it is atural for these poits to t i a triagular domai. Barycetric coordiates: Let p 1, p, p 3 be the vertices of a referece triagle ad p be a poit i the it is always possible to write p as a barycetric combiatio of p 1, p, p 3 as follows: p = up 1 + vp + wp 3 ; where (u; v; w) are the barycetric coordiates of p with respect to ad are give by the area ratios u = area(p; p ;p 3 ) area(p 1 ;p ;p 3 ) ; v= area(p 1;p;p 3 ) area(p 1 ;p ;p 3 ) ; w= area(p 1;p ;p) area(p 1 ;p ;p 3 ) ; el.: ; fax: address: rababah@just.edu.jo (A.Rababah) /03/$ - see frot matter c 003 Elsevier B.V. All rights reserved. doi: /s (03)0043-1

2 34 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) where area(p 1 ;p ;p 3 ) is the area of the triagle with vertices p 1 ;p ;p 3.t is clear that u; v; w 0 ad u + v + w =1. he geeralized Berstei polyomials: Usig the otatio =(i; j; k) t, U =(u; v; w) t ad = i + j + k, U = u + v + w the the geeralized Berstei polyomials of degree over a triagle are deed by B (U)= U =! i!j!k! ui v j w k ; = ; U =1: he geeralized Berstei polyomials of degree over a triagle make a partitio of uity B (U)=1: he last sum ivolves ( + 1)( +)= terms, ad B (U) 0 wheever U =(u; v; w) t 0: he product of two geeralized Berstei polyomials is also a geeralized Berstei polyomial ad give by + J B (U)BJ m (U)= B+J +m (U); (1) + m where the biomial coeciet of two vectors =(i 1 ;i ;i 3 ) ad J =(j 1 ;j ;j 3 ) is deed by i1 i i3 = : J j 1 j j 3 he geeralized Berstei polyomials also satisfy a recurrece relatio. he set {B (U)} of Berstei polyomials of degree form a basis for the space of polyomials of total degree over the referece triagle.cosider the parametric represetatio X (U)= b B (U) of a triagular Bezier surface of degree with the Bezier poits (cotrol poits) {b }.Applyig the de Casteljau algorithm shows that the triagular Bezier patch X (U) satises the covex hull property over i the sese that mi b 6 X (U) 6 max b.he problem of degree reductio is cocered with dig aother set of Bezier poits (cotrol poits) {c } 1 deig the approximative triagular Bezier surface of degree 1, Y (U)= 1 c B 1 (U); so that the least-squares distace d(x; Y ) betwee X ad Y is a miimum.he problem of degree reductio for Bezier curves has bee studied i [5].t is proved i [6,7] that the best L -approximatio is equivalet to the problem of dig the best Euclidea approximatio of the Berstei Bezier coeciets.for more o triagular Bezier surfaces, see [1,,4].

3 . Prelimiaries A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) First, the elemets of the set of Bezier poits {b } are ordered i the followig form: {b ;0;0 ; b 1;1;0 ;:::;b 0;;0 ;b 0; 1;1 ;:::;b 0;0; ;b 1;0; 1 ;:::;b 1;0;1 ;b ;1;1 ;:::;b 1; ;1 ;b 1; 3; ;:::;b 1;1; ; :::}. Example. he elemets of {b } =7 are ordered i the followig form: {b 7;0;0 ;b 6;1;0 ;b 5;;0 ;b 4;3;0 ;b 3;4;0 ; b ;5;0, b 1;6;0 ;b 0;7;0 ;b 0;6;1 ;b 0;5; ;b 0;4;3 ;b 0;3;4 ;b 0;;5 ;b 0;1;6 ;b 0;0;7 ;b 1;0;6 ;b ;0;5 ;b 3;0;4 ;b 4;0;3 ;b 5;0; ;b 6;0;1 ;b 5;1;1 ; b 4;;1 ;b 3;3;1 ;b ;4;1, b 1;5;1 ;b 1;4; ;b 1;3;3 ;b 1;;4 ;b 1;1;5 ;b ;1;4 ;b 3;1;3 ;b 4;1; ;b 3;; ;b ;3; ;b ;;3 }. he followig result gives the itegral of the geeralized Berstei polyomials. Lemma 1. he itegral of the geeralized Berstei polyomials over a triagular regio is give by B A (U)dA = ( + 1)( +) ; () where A is the area of the triagular regio. Proof. he proof is aalogous to the formula i [3, Sectio 5.7] by rst showig that b B (U) A da = b ( + 1)( +) ad ow the lemma follows. Lemma. he L orm of the triagular Bezier surface X (U)= b B (U) is give by X (U) A = + J b b J : (3) J = ( + 1)( +) Proof. X (U) = = = b B (U) da b b J B (U)BJ (U)dA J = b b J J = B (U)B J (U)dA

4 36 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) = b b J J = ( + J ) B +J (U)dA: vokig Lemma 1 completes the proof. Let r =( + 1)( +)= ad the r r matrix Q be give by [] A + J Q = : (4) ( + 1)( +) t is clear from the deitio that the matrix Q is a real symmetric matrix.usig the mathematical iductio, all the upper left submatrices of Q have positive determiats ad, thus, the matrix Q is a symmetric positive deite matrix, see [9].he sum of each colum ad each row of the real symmetric matrix Q equals to A=( + 1)( + ). hus, the L orm of X (U) is give i matrix form by X (U) = b t Q b; where the vector b cotais the elemets of the set of Bezier poits {b } ordered i the form described before.t is also clear from Eq.(5) that the matrix Q is positive deite. (5) Example. For the case of uit triagular regio with = the L X (U) = bt Q b where Q = ad b t =(b ;0;0 ;b 1;1;0 ;b 0;;0 ;b 0;1;1 ;b 0;0; ;b 1;0;1 ). orm of X (U) is give by 3. Degree raisig Give a triagular Bezier surface X (U) of degree with Bezier poits {b }, we wat to write the same triagular Bezier surface X (U) usig a basis of degree +1 with Bezier poits {b } +1; b B (U)= b B +1 (U): (6) +1

5 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) Multiplyig the left-had side by U = 1 ad comparig the coeciets of the same powers gives (see [3]) Let b = r 1 = i +1 b e 1 + j +1 b e + ( + 1)( +) ; r = k +1 b e 3 : (7) ( + )( +3) ; r 3 = he the degree raisig ca be writte i the matrix form b = b; ( 1) ; r 4 = ( 1)( ) : where the vector b cotais the r elemets with = + 1 ad the vector b cotais the r 1 elemets with = ordered as described before.he matrix has dimesio r r 1, ad ca be writte i the block form = 1 B 0 ; +1 C where block B has dimesio (3+3) (3), ad block 0 of zero elemets has dimesio (3+3) r 4 ad C has dimesio r 3 r 1.he matrix B has the form B = : he matrix C does ot have a regular costructio ad, thus, should be determied usig formula (7) of degree raisig.

6 38 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) he L orm Let X (U) ad Y (U) be triagular Bezier surfaces of degree ad + 1, respectively, with X (U)= b B (U); Y (U)= +1 c B +1 (U); where {b } ad {c } +1 are the correspodig Bezier poits.we cosider the L measure of distace betwee X (U) ad Y (U) as follows: d (X; Y )= X (U) Y (U) da = b B (U) +1 c B +1 (U) da: Usig the degree raisig i (7) weget d (X; Y )= = b B +1 (U) (b c )B (U) c B +1 da: (U) da Lettig d = b c, ad usig Lemma gives d (X; Y )= d d J B +1 1 = + +1 = d t Q +1 d; +1 J =+1 A ( + 3)( +4) (U)BJ +1 (U)dA +1 J =+1 + J d d J where the (( + )( +3)=) (( + )( +3)=) matrix Q +1 is deed i (4). hus, the L measure of distace betwee the triagular Bezier surfaces X (U) ad Y (U) of degree ad + 1, respectively, is give i the followig theorem.

7 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) heorem 3. he L measure of distace betwee the triagular Bezier surfaces X (U) ad Y (U) of degree ad +1, respectively, is give by d (X; Y )= d t Q +1 d; where d = b c. 5. Degree reductio this sectio, we use the L orm ad the discrete l orm to measure the distace of degree reductio betwee the triagular Bezier surfaces X ad Y. Give a set of Bezier poits {b } which dees the triagular Bezier surface of degree, X (U)= b B (U); we wat to d aother set of Bezier poits {c } 1 deig the approximative triagular Bezier surface of degree 1, Y (U)= c B 1 (U); 1 so that the least-squares distace d (X; Y )= d t Q d betwee {b } ad {c } 1 is miimized.substitutig d = b c, where c = c, ad doig some simplicatios i d (X; Y )weget d t Q d =(b c ) t Q (b c ) =(b 1 c) t Q (b 1 c) = b t Q b b t Q 1 c c t t 1Q b + c t t 1Q 1 c = b t Q b c t t 1Q b + c t t 1Q 1 c: We use the least-squares method to d c, see [8].For a miimum of d t Q d to occur, it is ecessary that the derivative of d t Q d with respect to the elemets of the vector c is zero.solvig the ormal equatios gives 0= 9 9c (dt Q d)=( t 1Q b + t 1Q 1 c) t 1Q 1 c = t 1Q b: Sice 1 t Q 1 = Q 1, ad the matrix Q 1 is a symmetric positive deite matrix, the matrix 1 t Q 1 is ivertible.hece ( 1 t Q 1 ) 1 exists, ad the least-squares distace is (8)

8 40 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) miimized by choosig c =( t 1Q 1 ) 1 t 1Q b: he matrix ( 1 t Q 1 ) 1 1 t Q is the pseudo-iverse (geeralized iverse) of the matrix 1, see [9, Sectio 13.7]. Substitutig the last formula for c i the least-squares distace d (X; Y )i(8) gives the error i the followig theorem. heorem 4. he error of the L measure of distace for the degree reductio satises = b t Q b b t Q 1 ( t 1Q 1 ) 1 t 1Q b: For the discrete l measure of distace of degree reductio we wat the least-squares distace d D (X; Y )= d t d betwee {b } ad {c } 1 to be miimized.after some calculatios, similar to the last case, we get d t d = b t b c t t 1b + c t t 1 1 c: Fidig ad solvig the ormal equatios gives the solutio c D =( t 1 1 ) 1 t 1b: he matrix ( 1 t 1) 1 1 t is the pseudo-iverse (geeralized iverse) of the matrix 1.Substitutig the last formula for c i the least-squares distace d D (X; Y )i(10) gives the error i the followig theorem. heorem 5. he error of the l measure of distace for the degree reductio satises = b t b b t 1 ( t 1 1 ) 1 t 1b: Eq.(9), which illustrates degree reductio with Bezier surfaces, is the Moore Perose iverse of the degree elevatio matrix 1.Eq.(11), which illustrates degree reductio with Bezier cotrol poits, is the Moore Perose iverse of the degree elevatio matrix 1.So c ad c D are the same solutio. (9) (10) (11) Ackowledgemets he author would like to thak the referee for useful commets. Refereces [1] C.de Boor, B-form basics, i: G.Fari (Ed.), Geometric Modelig: Algorithms ad New reds, SAM, Philadelphia, 1987, pp [] G.Fari, riagular Berstei Bezier patches, Comput.Aided Geom.Des.3 (1986)

9 A. Rababah / Joural of Computatioal ad Applied Mathematics 158 (003) [3] G.Fari, Curves ad Surfaces for Computer Aided Geometric Desig, Academic Press, Bosto, [4] J.Hoschek, D.Lasser, Fudametals of Computer Aided Geometric Desig, A.K.Peters, Wellesley, MA, [5] B.G.Lee, Y.Park, he distace for the Bezier curves ad degree reductio, Bull.Austral.Math.Soc.59 (1997) [6] D.Lutterkort, J.Peters, U.Reif, Polyomial degree reductio i the L -orm equals best Euclidea approximatio of Bezier coeciets, Comput.Aided Geom.Des.166 (1999) [7] J.Peters, U.Reif, Least squares approximatio of Bezier coeciets provides best degree reductio i the L -orm, J.Approx.heory 104 (000) [8] J.Rice, he Approximatio of Fuctios, Vol.1, Liear heory, Addiso-Wesley, Readig, MA, [9] C.Ueberhuber, Numerical Computatio, Methods, Software, ad Aalysis, Spriger, Berli, 1997.