This note generalizes a result of Goodman[3], where it is shown that the convexity of Bèzier nets defined on a base triangle is preserved on sub-triangles obtained from a mid-point subdivision process. Here we show that the convexity of Bèzier nets is preserved on and only on sub-triangles that are "parallel" to the base triangle

Gregory, JA

Zhou, J

English

Brunel University Research Archive

                     TR/04/90      March 1990  CONVEXITY OF BẺZIER NETS ON SUB-TRIANGLES  John A. Gregory and Jianwei Zhou      With minor revisions, November 1990.  To appear in CAGD.                   Convexity of Bẻzier Nets on Sub-triangles    by    John A. Gregory and Jianwei Zhou   Department of Mathematics and Statistics, Brunei University,  Uxbridge, UB8 3PH, England   Abstract: This note generalizes a result of Goodman[3], where it is shown          that the convexity of Bèzier nets defined on a base triangle is preserved on          sub-triangles obtained from a mid-point subdivision process. Here we show          that the convexity of Bèzier nets is preserved on and only on sub-triangles          that are "parallel" to the base triangle.   1.Introduction   Let T be a triangle, called the base triangle (see [1]), with vertices V1,V2andV3. (Here, and elsewhere in the paper, we assume that triangles are non-degenerate, that is, their vertices are not colinear.) Then each point P of the plane determined by V1,V2andV3.can be    represented by its barycentric coordinates (u,v,w) with respect to the base triangle T as  (1)  P = ,wVvVuV 321 ++   u+v+w = 1.   Denote fn as a set of (n+1)(n+2)/2 values  (2)  { }.0,k0,j0,inkjiRff ijkn ≥≥≥=++∈=   -1-           The Bernstein polynomial of fn over T is then given by   (3)                     ∑=++=nkjinijkijkn ,w)v,(u,Bfw)v,u,;B(f  Where (4)          kjinijk wvuk!j!i!n!w)v,(u,B =            are Bernstein basis functions. If we let shift operators E1, E2 and E3 with respect to T be defined by  (5)  ,ffE,ffE,ffE 1kj,i,ijk3k1,ji,ijk2kj,1,iijk1 +++ ===   then the Bernstein polynomial can be represented symbolically as  (6)   .f)wEvE(uEw)v,u,;B(f 000n321n ++= Now we consider (n+l)(n+2)/2 points of T with barycentric coordinates (i/n,j/n,k/n), namely,  .nkji,(k/n)V(j/n)V(i/n)VP 321ijk =++++=   Connecting     ,P,P,P 1kj,i,k1,ji,kj,1,i +++ we obtain a triangle, denoted  for i+j+k=n-l. Similarly, a triangle  is obtained for  ijkU ijkW i+j+k=n+l with   1kj,i,k1,ji,kj,1,i P,P,P −−−as its vertices. A piecewise linear function ( ) ( )nn fLor,wv,u,;fL  for short, is defined on T such that it satisfies  (7)  ( ) ,nkji,fP;fL ijkijkn =++=   and is linear on each triangle  We call .WorU ijkijk ( )wv,u,;fL n  the Bėzier net of  .fn The approximation theory of Bernstein polynomials and their applications in CAGD,  Computer-aided Geometric Design, indicate that the Bėzier net L  is closely related to  )f( nw)v,u,;B(fn  and reflects certain features of . As this note is concerned with the  )wv,u,;B(fnconvexity, we state the following results (see [1,2]):  -2-   Theorem 1   (i) If the Bėzier net L(fn;u,v,w) is convex with respect to T, so is the Bernstein  polynomial . w)v,u,;B(fn(ii) The Bėzier net  is convex with respect to T if and only if w)v,u,;L(fn (8)   ⎪⎩⎪⎨⎧+≥++≥++≥++++++++++++++++++++++,ffff,ffff,ffff1k1,ji,1kj,1,ik1,j1,i2kj,i,1k1,ji,k1,j1,i1kj,1,ik2,ji,1kj,1,ik1,j1,i1k1,ji,kj,2,i for i+j+k=n-2.  Using the shift operators, one may rewrite (8) as   (9)  ( )( ) ,2nkji,0fEEEE ijkiiii 3121 −=++≥−−   for any permutation { } { }.1,2,3ofi,i,i 321    be the vertices of another triangle  in the same plane as T and let *3*2*1 VandV,VLet*T (10)  P=  ,1wvu,VwVvVu ****3**2**1* =++++ define the barycentric coordinates ( )*** w,v,u  of P with respect to . Assume that   *T hasV*ibarycentric coordinates  with respect to T, that is, )w,v,(u iii (11)    1wvu,VwVvVuV iii3i2i1i*i =++++= for i=l, 2, 3. Then   (12)   ⎪⎩⎪⎨⎧++=++=++=.wwwvwuw,vwvvvuv,uwuvuuu3*2*1*3*2*1*3*2*1*  A Bernstein polynomial on is defined symbolically by *T (13)  ( ) ( ) ,fEwEvEuw,v,u;fB *000n*3**2**1*****n ++=   Where  (14)  { }0k0,j0,in,kjiRff *jki*n ≥≥≥=++∈=   and define the shift operators on  We then have: *iE .T* Theorem 2 (Chang and Davis[l]) Let (15)      ( ) ( ) ( ) ,fEwEvEuEwEvEuEwEvEuf 000k332313j322212i312111*ijk ++++++=   -3- for i+j+k=n. Then ( )****n w,v,u;fB  is the Bernstein representation of B( ;u,v,w) with respect to nf.T*    Proof:  From (12) and (15) we obtain, by equating coefficients,   ( ) ( ) 000n321n fwEvEuEwv,u,;fB ++=                ( ) ( ) ([ ] 000n332313*322212*312111* fEwEvEuwEwEvEuvEwEvEuu ++++++++= )   =  ( ) *000n*3**2**1* fEwEvEu ++ if and only if   ( ) ( ) ( ) .fEwEvEuEwEvEuEwEvEufEEE 000k332313j322212i312111*000k*3j*2i*1 ++++++++=    Remark In the above proof we note the identities  (16)   ,1,2,3i,EwEvEuE 3i2i1i*i =++= where, in symbolic manipulation involving these operators, indices in operator expressions must  sum to n and the operator expressions are applied to  .ff 000*000 =       The set determines a new Bezier net *nf ( )****n w,v,u;fL  which is a piecewise linear function  on  We call .T* ( )*nfL  the restricted Bezier net of L(fn) on  Naturally,  is called a  .T* *Tsub-triangle of T if  the vertices of  are all inside or on the boundary of T.  ,VandV,V *3*2*1*TIn this case  1,2,3.i0,wvu ii,i, =≥Let  be a sub-triangle of T. Then if *T ( )wv,u,;fB n  is convex with respect to T, so is  ( )****n w,v,u;fB with    respect to   One would ask whether or not a similar result holds for the Bėzier nets. Grandine[4] showed a negative result but in [3], Goodman shows that if  is a  .T**Tsub-triangle obtained from a "mid-point" subdivision process, then the convexity of L(fn) does guarantee  the convexity of the restricted Bėzier net ( )*nfL  with respect to . In the next section, we generalize this result and show that only a very limited class of sub-triangles have  *Tthe property of preserving the convexity of Bėzier nets.   -4-   2. Main Result   Let  be a non-degenerate triangle that lies on the plane determined by the base triangle *TT. We say is parallel to T if each of the edges of is parallel to one of those of T. We now  *T *Tmake the following definition:   Definition A non-degenerate sub-triangle  is called "Bėzier net convexity preserving" if, for  *Tall Bėzier nets L(fn) that are convex with respect to T, the restricted Bėzier nets ( )*nfL  on are  *Talso convex with respect to  *T  Noting the barycentric coordinate representation of the vertices of with respect to T in  *T(11), we have the following lemma.   Lemma The following statements are equivalent,  (i) is parallel to T.  *T(ii) There exist a non-zero scalar ρ  and a permutation { }321 i,i,i  of {1,2,3} such that  (17)   ).VV(VV),VV(VV),VV(VV 32*i*i13*i*i21*i*i 321321−=−−=−−=− ρρρ (iii) None of the sets { } { } { }321321321 w,w,wandv,v,v,u,u,u  has all distinct members.     Figure. Examples of parallel triangles   This lemma is illustrated by the two examples of parallel triangles labelled as shown in the  -5- figure. Property (ii) is simply a statement that the edges are parallel and property (iii) is a  restatement of this property in terms of the barycentric coordinates of     1,2,3.i,VwVvVuV 3i2i1i*i =++= Thus, in the examples .wwandvv,uu 211332 ===  . We now present our main theorem:  Theorem 3 Let  be a non-degenerate sub-triangle of T. Then is Bėzier net convexity  *T *Tpreserving if and only if it is parallel to T.   Proof: Suppose  is parallel to T and let L(f*T n) be any Bėzier net that is convex with respect to  T.  Comparing  (16)  and (11),  it  follows  by  statement  (ii) of the  lemma  that there exist a  non-zero scalar ρ  and a permutation { }321 i,i,i  of {1,2,3} such that  ).Eρ(EEE),Eρ(EEE),Eρ(EEE 32*i*i13*i*i21*i*i 321321−=−−=−−=−   Thus, by (16), we have for i+j+k=n-2         ( )( ) ( )( ) 000fEEEEEEEfEEEE **i*i*i*ik*3j*2i*1*ijk*i*i*i*i 32323222 −−=−−       = ( ) ( ) ( ) ( )( ) 0003121k332313j322212i3121112 fEEEEEwEvEuEwEvEuEwEvEuρ −−++++++       = ( ) ( ) ( )333kλµν222jαβγ111irst2kvµλitsr lyβαw,v,uBwv,uBw,v,uBρ∑∑ ∑=++=++ =++ ( )( ) νytµ,βδλ,αr3121 fEEEE ++++++−−       .0≥ Similarly,  ( )( ) ( )( ) .0fEEEE,0fEEEE *ijk*i*i*i*i*ijk*i*i*i*i 23133212 ≥−−≥−−    We therefore conclude that L ( )*nf  is convex with respect to T*.   W ow suppose that . is not parallel to T. By statement (iii) of the lemma, at least one  e nof the sets { }, and *T321 u,u,u { }321 v,v,v { }321 w,w,w  has all distinct members. Without loss of  generality, we assume  Then we have 0.uuu 312 ≥>> ( )( ) 0.uuuu 3121 <−−  Consider the situation where  -6-   (18)  ,nkji,f 1n,iijk =++δ= −  that is,  and                    f1fn,0,0 = ijk=0 if  1i −≤ . Obviously, the Bezier net ( )nfL  defined by (18) is convex with respect to T. Using (15) we obtain     nkji,uuuf k3j2i1*ijk =++= Hence for i+j+k=n-2 we have    ( )( ) ( )( )3121k3i2ii*ijk*3*1*2*1 uuuuuuufEEEE −−=−−   and, particularly for n  2, ≥ ( )( ) ( )( ) 0.uuuuufEEEE 31212n2* 2,0n0,*i*i*i*i 3121 <−−=−− −−   Thus the restricted Bėzier net ( )*nfL  is not convex with respect to T*.   Acknowledgements: This work was supported by the ACME Directorate of the Science and  Engineering Research Council and by Siemens AG.   References  [1] Chang, G. and P.J.Davis(1984), The convexity of Bernstein polynomials over triangles, J. Approximation Theory 40, 11-28. |2] Dahmen, W. and C.A.Micchelli(1985), Convexity of multivariate Bernstein olynomials and box spline surfaces, Research Report RC 11176, T.J.Watson Research Center.  [3]  Goodman, T.N.T.(1989), Convexity of Bėzier nets on triangulations, AA/899, Department of Mathematics and Computer Science, University of Dundee.  [4] Grandine, T.A.(1989), On convexity of piecewise polynomial functions on triangulations, Computer Aided Geometric Design 6, 181-187.  -7-   

Convexity of Bẻzier nets on sub-triangles

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