TIME-EFFICIENT NURBS CURVE EVALUATION ALGORITHMS

Transcription

1 TIME-EFFICIENT NURBS CURVE EVALUATION ALGORITHMS Kestuts Jankauskas Kaunas Unversty of Technology, Deartment of Multmeda Engneerng, Studentu st. 5, LT-5368 Kaunas, Lthuana, Abstract: Ths aer analyses tme-effcency of exstng NURBS evaluaton algorthms. The most comettve comutaton methods are modfed to acheve even better erformance. Performance tests ndcate that NURBS curve evaluaton tme-effcency can be mroved n unform and non-ratonal B- slne cases. Suggested otmzatons are very effectve n the evaluaton of hgher degree slnes wth a larger number of control onts. Keywords: NURBS, curve evaluaton, nverted trangular scheme Introducton NURBS stands for Non-Unform Ratonal B-Slne. It s the most oular slne reresentaton n today s commercal CAD ackages [, 4, 5, 8, ]. NURBS s able to reresent large varety of shaes, lke crcles, hyerbolas, arabolas, and stll reserves mathematcal exactness [5]. Generally, a slne s a smooth curve nterolated among gven control onts. Unfortunately, a slne cannot be constructed n the model sace drectly. Each ont on NURBS curve or surface must be calculated from the set of control onts, knot vector, and bass functon of secfc degree. Ths rocess s called NURBS evaluaton [, 5, 8]. In the followng sectons we wll dscuss theoretcal asects of NURBS as well as exstng evaluaton algorthms. Moreover, we wll ntroduce certan modfed evaluaton algorthms and strateges for unform and non-ratonal cases that mrove evaluaton erformance. Fnally we wll comare actual tme-effcency of suggested method mlementatons. 2 NURBS n Theory Acronym NURBS defnes secal roertes of ths artcular slne: () Non-Unform, (2) Ratonal, (3) B-Slne. Let us clarfy those roertes by startng from the last one. It has the bass functon of B-slne, whch ensures smooth blendng [2] of control ont nfluence over the curve. All theory regardng a regular B-slne s covered n Secton 2.. Ratonal roerty gves more flexblty to a slne [4, 5, ], but also ncreases comlexty. It s acheved by addng weghts to control onts. Ratonal B-slne s resented n Secton 2.2. Fnally, ntroducng the edtable knot vector to the slne concet, allows usage of non-unform ece-wse features [, ]. They are covered n Secton B-Slne Regular B-slne s defned by a set of control onts P, a knot vector U = { u j }, and degree, where =.. n, j =,.. m and m = n+ + [, 4, 6, 9]. Control onts are located n the mult-dmensonal sace we refer to as the model sace. The slne nterolates between control onts wth the hel of the bass functon: = < n = C( N u P, (), ( ) where s the degree of the bass functon, u s a coordnate n the arametrc slne sace and C ( s ont on curve n the model sace. Accordngly, any ont on the curve s obtaned by summng multlcatons of control onts P and bass functons N, (. The bass functon s calculated from the exresson: N f u u< u N u = +, ( ), otherwse (2) u u u u + + = N, ( + N+, ( u u u u ), (3), ( u where u s the coordnate n the arametrc slne sace and u j are values from the knot vector U. The last exresson s referred to as Cox-de Boor recurson formula [, 3]. It denotes that bass functon domans are dvded by elements of the knot vector,.e. knots [, 9]. It s also known that the sum of all bass functons N, ( equals one [7]. The sum of N, ( n the nterval k k equals one as well, where u k u< u k + (the artton of unty [ 6]): - 6 -

2 k = k N (. (4), = As the bass functon s non-negatve, t means that all bass functons N, ( outsde the nterval k k are zero. Consequently, all multlcatons N, ( P outsde the same nterval are zero. Thus, such control onts has no effect on the orton of the curve wthn u k u< u k +. So any ont on the curve C ( of the unform B-slne s affected by + control onts, wth the exceton of C () and C (). These secal cases can be exlaned through analyss of the knot vector. The knot vector s a set of non-decreasng values u j u, where j =,.. m and m = n+ +. In ths aer we use a normalzed knot vector form, so the arametrc sace of the B-slne and knot vector values are bounded by and. Also, t s a common ractce to use the clamed knot vector, where the frst + values equal and the last + values equal [, 3, ]: U = { u = u =... = u = u + u un un =.. = un+ = un+ + = }. (5) Let us examne the examle of a cubc B-slne ( = 3 ), defned by eght control onts ( n = 8 ) and m = 2 unform knots: U = {,,,,.. 4,. 6,. 8,,,, }. Bass functons are gven n Fgure 2. For u =.3 functons N,3 (.3 ), N 3 (.3 ), N 3,3 (.3 ) and N 4,3 (.3 ) are greater than zero, other functons are zero. Ths means that a ont on the curve C (.3) s affected by the oston of four control onts: P, P 2, P 3, and P 4. Also functon N 3 (.3 ) and N 3,3 (.3 ) values are sgnfcantly greater than values of N,3 (.3 ) and N 4,3 (.3). Ths suggests that the ont C (.3) s closer to P 2 and P 3 than to P and P 4. j+ In the case when u s n the oston of the knot u = u 5 =. 4 we have only three non-zero functons: N 3 (.3), N 3,3 (.3) and N 4,3 (.3). Consequently C (.4) s affected by three control onts: P 2, P 3, and P 4. So, a ont on the curve C ( s affected by s+ control onts, where s s knot multlcty s at u. Therefore the curve becomes C contnuous at ths ont (here the symbol C refers to slne contnuty and has a dfferent meanng than C (, see Secton 2.3 for more detals) [, 9]. Because of the clamed knot vector the frst curve ont C () s affected only by one control ont P. The last curve ont C () s affected by the control ont P resectvely: n C ( ) = P, (6) ( ) = P n C. (7) 2.2 Ratonal B-Slne Regular B-slne s qute owerful nterolaton tool, but t lacks flexblty. B-slne can not reresent conc sectons, lke crcles [4, 5, 9]. Therefore a ratonal form s used to cover these cases [4, ]: < n = < n C( = N ( w P, (8) N, ( w = where the weght w > s attached to every control ont., Fgure. A crcle reresented by four ratonal B-slnes - 6 -

3 Let us take a look at the examle of a crcle reresented as four ratonal B-slnes n Fgure. Each quarter of the crcle s constructed from searate ratonal quadratc B-slne defned by control ont sequences: P, P, }, P, P, }, P, P, }, P, P, }. The weghts of the frst and the last control onts n each { P2 { 2 3 P4 { 4 5 P6 { 6 7 P sequence are. The weght of the mddle control ont s 2 / 2 [4]. Greater weghts ull the curve towards the control ont and lesser weghts ush the curve away [, 4, ]. Naturally, regular B-slne s a secal case of ratonal B-slne when all weghts are equal to. 2.3 Non-unform Ratonal B-Slne The term of unformty s used to defne a relaton between the sequence of control onts and the arametrc slne sace. As mentoned n Secton 2., control ont nfluence over the curve s defned by bass functons and functon domans are dvded by knots [9]. Ths means that roerty of unformty s embedded nto the knot vector []. Untl now we consdered a knot vector to be clamed and unform: U = { u =... = u =... u =... un =... = un+ + = }, (9) n where + < n. Such a knot sequence dvdes whole arametrc sace nto unform ntervals. Each of ntervals contans + non-zero bass functons, thus the curve s affected by + control onts n ths nterval (see Secton 2.). In general case, knots can be dstrbuted n non-unform manner. However a knot sequence must be non-decreasng, as shown n the exresson (5). Let us take the examle of the knot vector U = {,,,,.. 4,. 6,. 8,,,, } and modfy t by settng u = u = u. 2 : U = {,,,,.... 8,,,, }. Knot multlcty of s= at u = = leaves only one non-zero functon at ths ont (see Fgure 3), whch suggest that C (.2) s affected by sngle control ont. Therefore, the curve goes through ths control ont: C (.2) = P3. In other words, the knot of s = multlcty s reduces curve contnuty at that knot by s [3, ]. In ths examle the curve becomes C C contnuous at u =. 2. Further ncrement of multlcty s ontless, because t excludes control onts from affectng the curve. NURBS s owerful enough to comose any shae. Recall the examle of the crcle n Fgure. It was reresented by four unform ratonal B-slnes. Knot multlcaton n the knot vector allows the constructon of such a shae from sngle quadratc NURBS curve. The same control onts wth weghts are used n the sequence P, P, P, P, P, P, P, P, }. The last control ont s the same as the frst to close the curve. { P Instead of multle control onts, multle knots are emloyed: U = {,,,.25,.25,.5,.5,.75,. 75,,, } [4]. Behavor of bass functons s dected n Fgure 4. Notce that every quarter of the crcle s reresented by sngle non-zero knot nterval and each quarter s ndeendent. Fgure 2. Bass functons of unform cubc B-slne defned by eght control onts Fgure 3. Bass functons of cubc B-slne wth knot multlcty of three at

4 Fgure 4. Bass functons of quadratc NURBS defned by the knot vector U={,,, /4, /4, 2/4, 2/4, 3/4, 3/4,,, } 3 NURBS evaluaton algorthms To reresent NURBS n the model sace (Cartesan mult-dmensonal sace) as a curve, the slne must be evaluated at multle u, where u. Accordng to the exresson (8) bass functons are necessary n order to do so. Several bass functon calculaton methods are covered n Secton 3.. Once bass functons are known, they can be used to determne a ont on the curve. The descrton of sngle ont evaluaton algorthms can be found n Secton 3.2. Fnally, entre NURBS curve evaluaton strateges are resented n Secton Bass functon As we already dscussed n Secton 2., calculaton of all bass functons s not necessary. There are only s+ non-zero bass functons at any u, where s the degree of the bass functon and s s knot multlcty at u. So we are to obtan all bass functons from N k, ( to N k, (, where u k u< u k Cox-de Boor recurson The most obvous soluton s to use a standard Cox-de Boor recurson formula, gven n the exresson (2) and (3). Although ths formula s smle to understand and easy to mlement, [6] and [9] sources state that t nvolves many unnecessary calculatons. Fgure 5 llustrates how N k, ( s obtaned. Fgure 5. Comutaton of non-zero bass functons Zero functons are marked n blue. They have no effect on hgher degree functons n successve teratons, because multlcaton by zero s zero (blue arrows). In the examle of U = {,,,,.. 4,. 6,. 8,,,, }, where = 3 and k = 4 ( u s n the nterval u 4 u< u5 ), the recursve formula returns non-zero values of N ), N ), N ), and N ). Accordngly to the,3 ( u exresson (3), to obtan N ) the algorthm calculates N ) and N ). To acqure second degree,3 ( u 3 ( u 3,3 ( u,2 ( u, ( u ( u, ( u 2 ( u functons, the recurson must obtan frst degree functons N ), N ) and N ), N ). Fnally, frst degree functons s calculated from zero degree functons: N ) s acqured from N ) and N ), 4,3 ( u ( u 3, ( u, ( u ( u N ) s acqured from N ) and N ), N ) s acqured from N ) and N ). Notce that ( u ( u 3, ( u 3, ( u 3, ( u 4, ( u N ) s calculated twce, so N ) as well as N ) s actually calculated three tmes. Moreover, only ( u ( u N ) s non-zero among all zero degree functons. 4, ( u 3, ( u Ths examle llustrates how Cox-de Boor recurson formula s overloaded wth unnecessary calculatons. Naturally, the evaluaton of hgher degree B-slne bass functons yelds even more unnecessary teratons. Also the exresson (3) s numercally unstable, because of / cases [5]. Another drawback s noted

5 n [2]. The recurson formula gves an ncorrect result when u =. The last ont on the curve s always C ( ) = {,, }. To overcome ths roblem, we smly use exressons (6) and (7) as secal cases, so C () and C () can be found wthout the calculaton of bass functons Inverted Trangular Scheme To avod unnecessary calculatons, authors n [6] resent the algorthm based ITS (nverted trangular scheme). It s gven as Bass_ITS functon n seudo code. It calculates functons from lower to hgher degree n contrast to the recursve algorthm. Also, they suggest rearrangement of the exresson (3) to remove oeraton dulcatons: N L j+ R j+ = N k j, ( + N k j+, ( ), () R + L R + L k j, ( u j j+ j+ j where L = u and R = u + u. () j u k + j Bass_ITS(k,,. N[] = 2. for (j = ; j <= ; j++) 2.. saved = 2.2. L[j] = u - knots[k + - j] 2.3. R[j] = knots[k + j] - u 2.4. for (r = ; r < j; r++) tm = N[r] / (R[r + ] + L[j - r]) N[r] = saved + R[r + ] * tm saved = L[j - r] * tm 2.5. N[j] = saved 3. return N Note that k should already be known, where k defnes the knot nterval n whch u resdes. Therefore, the method FndKnotSan (avalable n [6]) must be aled to determne k before the mlementaton of Bass_ITS Modfed Inverted Trangular Scheme We notced another relaton. Let the rght art of the sum n N, ( be equal A, then the left art of the sum n N, ( s always A. Based on ths observaton, we roose another modfcaton of the exresson (3): N j k = A ( N ( + ( A ) N ( ), (2), (,, +, +, u where A ( = ( u u ) /( u + u ) and k k. (3), The examle of non-zero cubc bass functon calculaton s gven n Table, followed by modfed ITS algorthms. As u value s fxed we omt the notaton of (. Table. Non-zero bass functon calculatons for cubc B-slne, usng a modfed ITS = = = 2 = 3 3 N A N k k 3,3 = ( k 3) k 2 k 2 N k 2,2 = ( Ak,2 ) Nk, N k 2,3 = Ak 3 Nk 2 + ( Ak,3 ) Nk, 2 k N k, = ( Ak,) N k, N k,2 = Ak,2 N k, + ( Ak,2 ) N k, N k,3 = Ak,3 N k,2 + ( Ak,3) N k, 2 k N k, = N k, = Ak,N k, N k, 2 = Ak,2Nk, N k, 3 = Ak,3Nk,2 j Bass_ITS(k,,. N[] = 2. for ( = ; <= ; ++) 2.. for (j = ; j >= ; j--) 2... A = (u - knots[k - j]) / (knots[k + - j] - knots[k - j]) tm = N[j] * A N[j + ] += N[j] - tm N[j] = tm 3. return N Bass_ITSU(k,,. N[] = 2. M = (u - knots[k])/(knots[k+]-knots[k]) 3. for ( = ; <= ; ++) 3.. for (j = ; j >= ; j--) 3... tm = N[j] * (M + j)/ N[j + ] += N[j] - tm N[j] = tm 4. return N These algorthms return bass functons n reversed order: from N k, ( to N k, (. Bass_ITS and Bass_ITS algorthms are sutable for any NURBS. Only few CAD and CAM alcatons allow edtng

6 the knot vector, because such modfcaton s not ntutve []. Hence, n many cases NURBS stays unform. From the exresson (9) t s obvous that every non-zero nterval n the knot vector equals /( n ). Let us resume that M = A = u u ) /( u + u ). It s easy to calculate that A k, = M /, Ak, = ( M + ) / Ak 2, = ( M + 2) / k, ( k k k. So, n case of the unform knot vector, the exresson (3) can be smlfed:, M + j Ak j, =. (4) Pluggng the exresson (3) nto the last row of Table ndcates that calculaton of a non-zero functon set uses knots from u k + to u k +. So the equaton (4) s vald when all knot ntervals from u k + to u + are equal. In the case of the clamed knot vector, the frst and last knot ntervals are zero. As the k frst unform nterval begns at u and the last unform nterval ends at u n, the exresson (4) can be used for all ntervals from u + to un. Ths means that the ITS algorthm can be wrtten as Bass_ITSU for all u k u< u k+, where: 2 k n. (5) 3.2 Sngle ont on curve Each of non-zero functons defnes how strongly a certan control ont affects a curve (see Secton 2.). Accordng to the exresson (8), the strength of the effect s also modfed by weghts of control onts (see Secton 2.2). In order to calculate C (, we requre a sum of all N, ( w P dvded by the sum of N ( w,, where k k and u k u< uk+. Followng algorthms calculate a ont on the curve, when bass functons are known. Thus GetPont should be used after Bass_ITS. Because of the nverted functon order n Bass_ITS and n Bass_ITSU, those algorthms should be followed by GetPont. GetPont(N, k). Nsum = 2. Cu = {,, } 3. for ( = ; <= ; ++) 3.. Nsum += N[] *= P[k - + ].Wegth 3.2. Cu += N[] * P[k - + ].To3D() 4. return Cu/Nsum GetPont(N, k). Nsum = 2. Cu = {,, } 3. for ( = ; <= ; ++) 3.. Nsum += N[] *= P[k - ].Wegth 3.2. Cu += N[] * P[k - ].To3D() 4. return Cu/Nsum The method To3D() returns { x, y, z} coordnates and gnores the control ont s weght. If a slne s regular B-slne and all weghts equal, we can use the exresson () nstead of the exresson (8) to fnd a certan ont on the curve. In such case GetPont algorthm can be smlfed to GetPont_NR: GetPont_NR(N, k). Cu = {,, } 2. for ( = ; <= ; ++) 2.. result += N[] * P[k - ].To3D() 3. return Cu 3.2. De Boor s algorthm There are several B-slne evaluaton technques that do not need bass functons to determne a ont on the curve, lke de Boor s algorthm [9]. De Boor s algorthm s based on observaton that C ( s ostoned at the locaton of the control ont Pk, when u= uk and knot multlcty at u equals (see secton 2.3). How do we make desred knot multlcty at any u? The author n [9] suggests a multle nserton of a knot at u. The nserton of an addtonal knot also means the nserton of a new control ont, thus after teratons the last control ont s exactly at the oston of C (. In case when u s already at the oston of the knot u k wth multlcty s, only s teratons of the nserton are requred. The oston of every new control ont can be found from exressons [3, 9]: where a w w = a ) P w P Q ( + a, (6) u u = u u + for all k + k. (7) However, the actual nserton of knots s not erformed, because ths would lead to the modfcaton of the control ont sequence durng the evaluaton. Thus the sequence of new control onts s rocessed n a

7 temorary array. The exresson (6) requres control onts to be converted to a homogenous 4D coordnate w system by multlyng coordnates by weght: P = { w x, w y, w z, w }. Ths task s erformed by ConvertTo4D() method. The converson back to Cartesan 3D coordnate system s erformed by dvdng coordnates by weght P = x / w, y / w, z / w, w } n ConvertTo3D() method. { GetPont_DeBoor(k,. s = 2. whle (k >= s && knots[k - s] == 2.. s++ 3. Q = new ControlPont[ - s + ] 4. for ( = k - ; <= k - s; ++) 4.. Q[ - k + ] = P[].ConverTo4D() 5. for (r = ; r <= - s; r++) 5.. for ( = k - s; >= k - + r; --) 5... a = (u - knots[]) / (knots[ + - r + ] - knots[]) j = - k Q[j] = ( - a) * Q[j - ] + a * Q[j] 6. return Q[-s].ConvertTo3D().To3D() 3.3 Multle Ponts on Curve Generally, evaluaton of multle onts can be done usng sngle ont evaluaton several tmes. But several otmzatons can be made. To evaluate entre NURBS curve, we must obtan multle onts C (, where u=, u, 2 u... u, and u =/( stes ) s the ste n the arametrc slne sace. Under these condtons the ntal knot nterval s u u< u +, thus ntal k =. Successve k values can be traced easly, so the rocedure FndKnotSan n not needed. Also u = and u = are handled as secal cases (see Secton 3.) and calculated from exressons (6) and (7). The followng algorthm evaluates the number of onts equal to stes on any NURBS curve. NURBS_ITS(stes). ste = / (stes - ) 2. Cu = new Pont[stes] 3. Cu[] = P[].To3D() 4. ter = 5. u = knots[] + ste 6. for (k = ; k < n; k++) 6.. whle (knots[k] == knots[k + ] && knots[k] < ) 6... k whle (u < knots[k + ]) N = Bass_ITS(k,, Cu[ter] = GetPont(N, k) ter u += ste 7. C[stes - ] = P[n - ].To3D() 8. return Cu Algorthms n stes 6.2. and can be relaced by modfed Bass_ITS and GetPont resectvely. If a slne s known to be non-ratonal then GetPont_NR can be used n ste If a slne s unform t s ossble to otmze ths algorthm even further Evaluaton of Unform B-slne Curve Recall Secton 3..3 and exressons (5), whch states that Bass_ITSU can be used nstead of Bass_ITS wthn bounds of 2 k n. Fgure 6 llustrates the bass functons of the cubc unform B- slne defned by m = 7 knots. Notce that N ) = (), N.5) = (.95),,3 ( N 3,3 ( N, 3 N.) = (.9) and so on. Clearly, certan bass functons of the unform B-slne are symmetrcal to 3( N, 3 each other. Actually, any functon N, ( can be reflected to N n, ( at the mddle ont of the arametrc sace. We refer to ths oeraton as to ref : ref : N, ( N n, (, (8) where k k. (9)

8 The set of non-zero functons N k, ( N k, ( at u u < <. 5 can be cloned to k u k + N n k, ( N n k+, (. In other words, there s no need to calculate non-zero functons for the second half of the arametrc sace, because they can be obtaned from the frst one. Fgure 6. Bass functons of cubc unform B-slne defned by 3 control onts (7 knots) Fgure 6 also dects another mortant roerty of unform B-slne. Pay attenton to functons marked as red, they are dentcal: N.2) = N (.3) = N (.4) =... = (.8). The set of non-zero functons at 3,3 ( 4,3 5,3 N9, 3 u =.22 conssts of four functons: N 3 (.22 ), N 3,3 (.22 ), N 4,3 (.22 ), and N 5,3 (.22 ). There s a set of functons wth the same values at each nterval u k, where 5 k : N 3 (.22 ) = N 3,3 (.32 ) =... = N 7,3 (.72), N 3,3 (.22 ) = N 4,3 (.32 ) =... = N 8,3 (.72 ), N 4,3 (.22 ) = N 5,3 (.32 ) =... = N 9,3 (.72 ), and N 5,3 (.22) = N 6,3 (.32 ) =... = N,3 (.72 ). Obvously, non-zero functons at arbtrary u2 u< u2 can be reeated at u j ( n ) + /, where j ( n ) (2 ). In ths aer we refer to ths oeraton as to re : re : N, j, ( N + ( u+ j /( n )), (2) where j n 3+ for all < 2. (2) However, u values must be dstrbuted n secfc manner, n order to ht a requred u or u+ j /( n ). Ths means that the chosen ste u must dvde each non-zero knot nterval nto the same number of equal subntervals. If we consder that κ Ν s a natural number, then ste u must satsfy: u=. (22) ( n )κ Multle Pont on Curve Evaluaton Strateges We suggest several NURBS evaluatons strateges regardng gven observatons n Table 2. The strategy name corresonds to the case of the slne and to the bass functon algorthm. The nterval row ndcates the bounds of the arametrc sace for bass clone oeratons that are gven n the last table row (see exressons (8), (9), (2), and (2)). Table 2. NURBS curve evaluaton strateges Strategy Case Interval Bass method Get ont method Bass clone oeraton NURBS_ITS General u Bass_ITS GetPont NURBS_ITS General u Bass_ITS GetPont URBS_ITS URBS_ITS+U UBS_ITS+U Unform Unform n 3 Unform Non-ratonal n 3 u <.5 Bass_ITS GetPont N ref ( N ( )) u =.5 Bass_ITS GetPont < u2 n, ( =, u u Bass_ITS GetPont N ref ( N ( )) n, ( =, u u2 u< u2 Bass_ITSU GetPont N + j, ( u+ j /( n )) = re( N, ( ) < u2 u Bass_ITS GetPont_NR N ref ( N ( )) n, ( =, u u2 u< u2 Bass_ITSU GetPont_NR N + j, ( u+ j /( n )) = re( N, ( ) Note that n order to aly unform case otmzatons, the ste sze u must be set accordngly to the exresson (22) before evaluaton. An unform non-ratonal slne s evaluated usng the smlfed method GetPont_NR nstead of GetPont (exresson () nstead of (8))

9 4 Results Algorthms gven n Secton 3 were mlemented usng C# rogrammng language and.net framework. The erformance tests were acqured on Intel Core2 Duo.86 GHz x2 CPU, 3.GB RAM machne. The evaluaton of sngle ont or functon takes only few nanoseconds. Ths makes the comarson of evaluaton tme-effectveness hardly ossble. Therefore all evaluaton algorthms were aled 5 tmes at dfferent u. Ths rocedure was erformed several tmes and average calculaton tmes were recorded. Recursve Cox-de Boor, Bass_ITS, Bass_ITS, and Bass_ITSU bass functon evaluaton algorthms were tested on the unform 27 control ont B-slne. The same algorthms and de Boor s knot nserton method were emloyed to determne a sngle ont on the curve. Calculaton tmes are gven n Fgure 7 and Fgure 8 resectvely. Mlseconds Recursve ITS ITS ITSU n = 27 Recursve ITS ITS ITSU Degree Fgure 7. NURBS bass functon calculaton tmes n mllseconds Mlseconds Recursve DeBoor ITS ITS ITSU n = 27 Recursve DeBoor ITS ITS ITSU Degree Fgure 8. NURBS sngle ont on curve evaluaton tmes n mllseconds Calculaton tme of the recursve Cox-de Boor algorthm grows radly for every successve degree of B-slne. However, the ITS s notceably less affected by the degree ncrement. De Boor s knot nserton should be fast, because t has no bass functon calculaton hase. Accordng to Fgure 8, GetPont_DeBoor overtakes recursve algorthm only when 4, but s left far behnd by the ITS. Ths haens because of a large number of scalar multlcatons and conversons from 3D to 4D and back. The ITS takes less tme n the bass functon determnaton hase than n the oston acquston hase even when = 8. Therefore erformances of sngle ont evaluaton usng Bass_ITS, Bass_ITS or Bass_ITSU are very smlar. Due to oor erformance of recursve Cox-de Boor and de Boor s knot nserton algorthms they were not ncluded n multle ont evaluaton. The evaluaton of multle onts over entre 27 control ont NURBS curve was carred out usng strateges gven n Secton Results are gven n Fgure 9. The same strateges were aled to n = 8 control ont and n = 9 control ont curves. Performance atterns reman the same as n Fgure 9, but URBS_ITS+U and UBS_ITS+U rovded less tme economy. In these cases, less control onts mean fewer ntervals where the oeraton re can be aled (see exressons (2) and (2))

10 Mlseconds NURBS_ITS NURBS_ITS URBS_ITS URBS_ITS+U UBS_ITS+U Degree n = 27 NURBS_ ITS NURBS_ ITS URBS_ ITS URBS_ ITS+U UBS_ ITS+U Fgure 9. NURBS multle onts on curve evaluaton tmes n mllseconds The mlementaton of ref and re oeratons n evaluaton of unform ratonal B-slne saved from 3.7% to 24.6% of calculaton tme (comare URBS_ITS+U and NURBS_ITS). The algorthm desgned for unform non-ratonal B-slne saved from 2.4% to 32.8% of calculaton tme. Also, URBS_ITS+U and UBS_ITS+U were resectvely u to 8.7% and 27.3% more tme-effcent n comarson to NURBS_ITS. The evaluaton of hgher degree bass functons takes longer. In these cases ref and re oeratons can save more tme (comare URBS_ITS+U and NURBS_ITS n Fgure 9). There s one more fact to be taken nto consderaton. The ercentage of saved calculaton tme deends on the number of NURBS control onts. Accordngly to the exresson (2), there are n 3 + knot ntervals where re oeraton can be aled. If n < 3, ths otmzaton can not be mlemented even f B-slne s unform. 5 Conclusons In ths aer we analyzed three already known NURBS evaluaton algorthms. Test results showed that the recursve Cox-de Boor formula s hghly neffectve esecally n the evaluaton of hgher degree slnes. Although de Boor s knot nserton method erformed better whle evaluatng slnes of the fourth and hgher degree, t was sgnfcantly overtaken by nverted trangular scheme n all cases. Due to ths dscovery we comosed several modfcatons of the nverted trangular scheme and few evaluaton strateges desgned for secal cases of NURBS. Accordngly to the test results, the resented strateges saved u to 24.6% of evaluaton tme n the case of unform B-slne, and u to 32.8% n the case of unform non-ratonal B-slne. A sgnfcant gan of erformance was observed durng NURBS evaluaton of the degree > 4 wth the number of control onts greater or equal to 3. The stated facts lead to a concluson that tme-effcency of NURBS curve evaluaton based on the nverted trangular scheme can be mroved. Ths s acheved by recognzng unform and non-ratonal cases and mlementng evaluaton strateges resented n ths aer. Otmzatons are esecally effectve n the evaluaton of hgher degree slnes wth a larger number of control onts. References [] Andersson F., Bert K. Bezer and B-slne Technology. Master of Scence Thess. 23. [2] Deng J. J. Theory of a B-Slne Bass Functon. Internatonal Journal of Comuter Mathematcs. 23, volume 8, ssue 3, ages [3] Farn G. Curves and Surfaces for CAGD: A Practcal Gude 5th Edton. Morgan Kaufman Publshers. 2 ages [4] Fsher J., Lowther J., Shene C. K. If You Know B-Slnes Well, You Also Know NURBS!. ACM SIGCSE Bulletn. 24, volume 36, ssue, ages [5] Krshnamurthy A., Khardekar R., McMans S. Drect Evaluaton of NURBS Curves and Surfaces on the GPU. Proceedngs of the 27 ACM Symosum on Sold and Physcal Modelng. 27, ages [6] Pegl L., Tller W. The NURBS Book 2nd Edton. Srnger. 999, ages [7] Plukas K. Skatna metoda r algortma, Naujas Lankas. ages [8] Sánchez H., Moreno A., Oyarzun D., García-Alonso A. Evaluaton of NURBS surfaces: an overvew based on runtme effcency. WSCG SHORT Communcaton Paers Proceedngs, Scence Press. 24, ages [9] Shene C. K. CS362 Introducton to Comutng wth Geometry Notes Onlne access: htt:// [] Xe H., Qn H. Automatc Knot Determnaton of NURBS for Interactve Geometrc Desgn. Proceedngs of the Internatonal Conference on Shae Modelng & Alcatons. ages