Rapid Termination Evaluation for Recursive Subdivision of Bezier Curves

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1 Rapid Terminaion Evaluaion for Recursive Subdivision of Bezier Curves Thomas F. Hain School of Compuer and Informaion Sciences, Universiy of Souh Alabama, Mobile, AL, U.S.A. Absrac Bézier curve flaening by recursive subdivision requires ha he maximum excursion of he subdivided curve segmen be known so ha recursion can be erminaed once his value drops below he specified flaness crierion. A much more accurae mehod han he mos commonly used echniques o evaluae his disance is presened. This mehod sops recursion sooner, significanly reducing he number of generaed sraigh line segmens ha are used o approximae he curve o wihin he given flaness. The incremenal compuaional overhead is minimal. Key words: recursive subdivision, Bézier curves, flaening. Inroducion The fas graphical rendering of curves usually involves he reducion of he curve o an approximaing polyline a process called flaening he curve. The required goodness of fi is specified by a single scalar parameer, d fla, called he flaness. The mahemaical curve should no deviae from he sraigh-line segmens of he approximaing polyline by a disance greaer han he flaness. In he case of wo-dimensional cubic Bézier curves, flaening can be achieved by forward differencing [], or by recursive subdivision []. The firs echnique, while fas, has he disadvanage ha he number of line segmens in he generaed polyline is fixed, raher han depending on he curvaure and size of he curve being rendered. Aemps a adaping he sep size in forward differencing (by doubling or halving he sep size along he pah) are described in []. Recursive subdivision requires recursively bi- This research was funded by a gran from QMS Inc., Mobile, Alabama. secing he curve unil each of he resuling Bézier curve segmens lies compleely wihin a specified disance of he line segmen joining is endpoins, and hen replacing each of he resuling Bezier curves by he line segmen joining is endpoins. The sopping crierion for he recursion requires a compuaionally expensive calculaion of he acual maximum perpendicular disance, d max, ha a cubic Bézier curve, specified by is conrol poins P0,!, P, deviaes from he line hrough PP 0. Alernaively, a reasonably good bu inexpensive esimae of d max, denoed by d " max, such ha d " max dmax, is desired. We may sop he recursion when d max d fla, or, because of he previous requiremen, when d " max d fla. A mehod commonly used o esimae he maximum deviaion of he curve from he line hrough PP 0 is o calculae eiher he x- or y-componen of he vecor from I, he poin closes o conrol poin P on line hrough PP 0, o P (see Figure a similar figure could be drawn wih boh inner conrol poins being on he same side of he chord), depending on wheher he magniude of he slope of PP o is greaer or less han one respecively. The appropriae componen of I P is also calculaed. This forms he esimae d " max for his mehod; i.e., if eiher of hese componens is greaer han he flaness, he curve is subdivided, and each resuling Bézier segmen is recursively flaened. This mehod gives a good esimae only if. he magniude of he slope of PP o is close o one, and. he wo conrol poins are equidisan from he line PP o, and on he same side. The firs condiion will no be rue in general, bu he second will end o be rue when he subdivi-

2 sion has occurred ofen enough o be close o recursion erminaion. Here d max can be calculaed as 075. max( A, B ), which is no oo far from he esimaed value of X max( A, B). However, he esimae is abou 4% oo high if he slope of PP o is eiher verical or horizonal, and nearly 50% oo large in ha case if A - B. The excessively conservaive esimae by his echnique causes unnecessary recursion deph, and consequenly more line segmens/verices are generaed han would be necessary o provide a curve flaened o he given flaness. A final problem wih mos rendering implemenaions is ha if A and B are wihin he flaness crierion, bu I or I are ouside he line segmen PP o, recursion is coninued unil I and I are wihin a disance of P 0 or P equal o he flaness. Thus, a series of verices are generaed ha are collinear, and essenially redundan. However, his is done o ensure ha he curve undergoes he appropriae excursion in he direcion of PP o. Calculaion of maximum excursion In his work, an expression for d max, he maximum deviaion of he curve from he line hrough PP 0, in erms of A and B is developed. I is shown ha his expression can be approximaed o wihin.% of he acual values by a quadraic E in, where E min{ A, B}, and F F max{ A, B}. In paricular, he approximaing expression is more accurae a more criical values of E. This solves he firs (and major) problem. Using his approach, ess have shown ha F he number of verices generaed during Bézier drawing is reduced by abou 6%. PP o, has been solved by reducing he problem ino a number of cases, and solving for he exremum poin or poins on he curve explicily. These exremum poins are resolved o verices wihou he need for any recursion. Mahemaical derivaions We are given he four conrol poins of a Bézier curve, P0,!, P. We will ake closed Bézier curves (where he firs and las conrol poins a coinciden, i.e., P0 ª P) as a special case, by uncondiionally recursing once. Consider a normalized coordinae sysem wih s- and r-axes such ha he origin is a P 0 and he r-axis is in he direcion PP 0, and he s-axis is orhogonal o his, in a righ-handed sense. The conrol poin P should be a (,0). I should be noed ha orhogonal componen cubic equaions of Bézier curves are compleely separable. The normalized coordinaes of an arbirary poin Pc (,) r s in his new sysem are calculaed as follows, ( P0 - P P P P P P P y c )( y 0 - y )-( y 0 - x c )( x - x 0 ) x r L ( P0 - P P P P P P P y c )( y x 0 ) ( x 0x c )( - ) x y 0y s L where, L P0P ( P - P + P - P x 0 ) ( x y 0 ) y Noe ha he denominaor of he equaion for s can be omied since we will be using a funcion of he raio of wo s-values (L cancels ou) o evaluae a normalized (no involving L) maximum excursion, l norm, of he curve from he sraigh line hrough PP 0. The second problem, ha of generaing many redundan collinear verices when he curve is close o he line defined by P 0 and P, bu when he convex hull exends o beyond he limis of

3 P P A y A M M d max I B B x B y A x I P P 0 To ge acual disances from hese coordinaes, we mus muliply by L. When i comes o he flaness decision, we can compare ( lnorm L) lnorm L wih l fla (where l fla is he flaness crierion disance), hereby avoiding a square roo evaluaion. chord is independen of he order of he inner. Transverse componen of curve conrol poins P, and P, we can swap he s- The general equaion for he s-componen of coordinaes of P and P such ha i. Tha is, poins on he curve as a funcion of he parameric value, where 0, is ds s, and s v, where - v +. Thus, v-9 v s ( ) ( - ) s0 + ( - ) s + ( - s ) + s d () + ( 6v- ) + 9( -v) Wih he seleced coordinae sysem, s0 s 0 (i.e., curve endpoins lie on he r-axis), so 0 We now solve he equaion s () ( - + ) s + ( -s ) ( ) s + ( - ) s + ( v- ) + ( - v) 0 We wan o find maximum (in fac, here may be for, wo local maxima) excursion of he curve from he r-axis. A hese poins, ds d Figure Bézier wih P and P on opposie side of PP 0. ( ) s + ( 6-9 ) s 0 Le us rescale he s-axis such ha si, where s i max ( s,s ), and i,. Tha is, we rescale he s-axis so ha he conrol poin furhes from he line hrough PP 0 (he chord) is a disance + (on he lef of he chord). Since he maximum excursion of he curve away from he

4 v v v / ( - ) ± 4 / ( - ) - 4 / ( - ) / ( -v) ( - v) ± r - 4v v ( - v) ( - v) ± v - v+ ( - v) The maximum excursion of he curve will occur nearer o he P 0 end (where 0) since we had arranged ha s s. Thus, he soluion for ha we are ineresed in is, ( -v) - v - v+ max ( - v) Now, we can subsiue his value of back ino equaion () o ge he normalized (scaled) disance of he poin on he curve of maximum excursion from he line hrough PP 0. d () v s( (), v ) v + norm max To avoid he compuaional overhead of evaluaing his funcion, he funcion was empirically approximaed by a quadraic, ~ dnorm ( v) ( v ) v The consans were derived such ha max{ d ~ norm( v) -dnorm( v) - v + } is minimized, and d ~ norm( v) -dnorm( v) 0, - v +, and ~ dnorm( v) dnorm( v) a v. The consans were chosen such ha ha d ~ norm ( v) is a good esimae (he proporional error is never greaer han.55%), and never an underesimae. In pracical Bézier curves, where he curvaure is much larger han he flaness crierion, he value of v a he deph in he recursive subdivision when flaness has almos been achieved, he expeced value of v will be close o. This is where he esimaing funcion is mos accurae. The maximum excursions (boh acual and esimaed) of he curve from a sraigh line are obained by dividing hese disances by he magniude of he scale facor used above, ~ ~ d dnormsi d dnormsi alhough his evaluaion would no acually be needed in any implemenaion.. Longiudinal componen of he curve If he curve is fla wihin he given olerance, we may sill need o find he poins of maximum excursion of he curve in he longiudinal (i.e., r- axis) direcion (see Figure he s-axis direcion has been scaled up for he sake of clariy; P max is projeced ono PP 0, he r-axis). This siuaion is usually handled by coninued muliple recursions. We will call he poin of maximum excursion on he opposie side of P 0 from P (if such a poin exiss) he minimum poin, P min, and he poin on he opposie side of P from P 0 (if such a poin exiss) he maximum poin, P max. We need o find eiher (or boh) urning poins if hey occur ouside he segmen PP 0 since he approximaing polylines mus pass hrough hem. If boh hese poins are inside PP 0, hen he (fla) curve is covered by simply joining P 0 and P direcly. Thus, he flaened muliline is passes hrough a sequence of verices which may be one of ( P 0, P ), ( P 0, P min, P ), ( P 0, P max, P ) (e.g., Figure ), ( P 0, P min, P max, P ), or ( P 0, P max, P min, P ) (see below.) The general parameric equaion of he r- componen of a Bézier curve is, 4

5 P P max P P s-axis r-axis P0 P0 Figure Longiudinal curve approximaed by line segmens PP 0 max and Pmax P (Think of all poins projeced ono line hrough P 0 P.) r ( ) ( - ) r0 + ( - ) r + ( - r ) + r r + ( r - r) + ( r - r + r ) + ( r - r + r -r) dr d ( r - r ) + 6( r - r + r ) + ( r - r + r -r ) ( r- r0) + ( r0 - r+ r) + ( r- r + r-r0) 0 r r r r r r r r r r r r - ( ) ± ( ) - ( - 0)( ) ( y- y+ y-y0) r r r r r r r r r r r r - ( ) ± ( - ) + ( - 0) - ( - 0) ( r - r) + r -r 0 Since he r-coordinae as defined above is normalized so ha r 0 a P 0 and r a P, i.e., r 0 0, and r, hen min max ( r - r ) + r ( r -r - ) + r ( r - r ) + ( r -r )- r ( r -r - ) + r ( r - r ) + () These values of can be subsiued ino r ( ) ( - ) r0 + ( - ) r + ( - r ) + r (( -)( r-( r- r )) + ) wih r0 0, and r o yield r min and r max. The coordinaes of he urning poins (if hey exis) are hen P P + ( P - P ) r, P + ( P - P ) r e e min 0x x 0x min 0y y 0y min P P + ( P - P ) r, P + ( P - P ) r max 0x x 0x max 0y y 0y max j j 5

6 The condiion ha P min and P max exis and should be added are as follows: if (r r ) { if (r < 0) add P min if (r > ) add P max } else { if (r < 0) add P min else if (r > ) add P max else if (discriminaor > 0) { if (q max > ) add P max if (q min < 0) add P min } } References [] Foley, J,D., A. van Dam, S.K. Feiner, and J.F. Hughes, Compuer Graphics Principles and Pracice, Addison Wesley, 995. [] Lane, J., and L. Riesenfeld, A heoreical developmen for he compuer generaion of piecewise polynomial surfaces, IEEE Transacions on Paern Analysis and Machine Inelligence, PAMI-(), Jan 980, [] Shanz, M., and S. Lien, Rendering rimmed NURBS wih adapive forward differencing SIGGRAPH 89, where r, are he normalized r-coordinaes of he inner conrol poins, q min,max are he normalized r-coordinaes of he urning poins, and discriminaor is he expression under he square roo in equaion (). 4 Conclusion. In conclusion, he new mehod produces an average of 6% fewer verices for Bézier segmens (measured over a very large es-suie of Pos- Scrip pages conaining ypical vecor graphics), while mainaining he prescribed flaness crierion. Curves whose convex hull is narrower han he flaness disance bu longer han PP o are no recursively generaed (producing redundan collinear line segmens), bu exremum verices are explicily calculaed. The flaening procedure is faser overall since he number of generaed verices in he flaened curve approximaion is smaller. The ime o fill areas bounded by Bézier segmens will be also reduced since he number of verices is smaller. Finally, he size of he display lis is concomianly reduced, saving memory resources. 6