Study on Bezier Curve Variable Step-length Algorithm

Transcription

1 Available olie at Physics Procedia 25 (2012 ) Iteratioal Coferece o Solid State Devices ad Materials Sciece Study o Bezier Curve Variable Step-legth Algorithm Guorog Xiao a, Xuemiao Xu b * a Departmet of Computer Sciece ad Techology GuagDog Uiversity of Fiace Guagzhou, Chia b School of Computer Sciece ad Egieerig South Chia Uiversity of Techology Guagzhou, Chia Abstract Bessel Curve also kow as Bezier Curve, through which the geeral vector graphics software accurately draws curves, as the Bezier curve is relatively easy to calculate ad its stable characteristic, i may areas it has bee widely applied. However, for the deficiecy of less efficiecy exists i the commo Bezier Curve Geeratio Algorithm, the selectio of the parameter step-legth sigificatly affects the accuracy ad efficiecy of the geerated curve. This paper aims at the efficiecy available i the existig Bezier Curve Geeratio Algorithm to propose the Variable Step-legth Algorithm; by chagig the parameter step-legth of the curve geeratio algorithm, it ca sigificatly reduce the calculatios of a large umber of duplicate poits i the poit-by-poit geeratio algorithm; the algorithm ot oly maitais a higher accuracy, but also sigificatly improves the efficiecy geerated i the curve, with better applicatio Published by Elsevier Ltd. B.V. Selectio ad/or peer-review uder resposibility of of [ame Garry orgaizer] Lee Ope access uder CC BY-NC-ND licese. Keywords:Bezier Curve, Berstei Polyomial, Curve Geeratio Algorithm, variable step-legth 1. Itroductio Bessel Curve also kow as Bezier Curve, through which the geeral vector graphics software accurately draws curves, cosists of lie segmets ad odes; the ode is a draggable fulcrum, the lie segmet like a scalable rubber bad; the see pe tool i the drawig tools is used to draw such vector curves. Certaily there are also Bezier Curve tools i some comparative mature bitmap software, like PhotoShop ad so o. I Flash4, there is still o complete curve tool, which has bee provided i the Flash5. Bezier Curve is a mathematical curve used i two-dimesioal graphics applicatio programs. There are four poits i the defiitio of the curve: the startig poit, the edig poit (also kow as the achor poit), ad two separate itermediate poits. Slide the two itermediate poits, the shape of the Bezier Curve will chage. I the late sixties of the twetieth cetury, Pierre Bézier has applied mathematical methods to depict the Bezier Curve for the Reault's automobile idustry. The Bezier Curve i accordace with coordiates of ay poit i four positios draws out a smooth curve. Historically, people who studied the Bezier Curve have desiged such vector curve drawig method iitially i Published by Elsevier B.V. Selectio ad/or peer-review uder resposibility of Garry Lee Ope access uder CC BY-NC-ND licese. doi: /j.phpro

2 1782 Guorog Xiao ad Xuemiao Xu / Physics Procedia 25 ( 2012 ) accordace with the idea that determies the four poits with the kow curve parameter equatio. The iterestig poit of the Bezier Curve lies more i its "Rubber Bad Effect" ~ that is, with the regular move of poits, the curve will geerate chages like stretches of rubber bads, brigig the visual impact. I 1962, the Frech mathematicia Pierre Bézier firstly studied the method to draw such vector curves, ad proposed a detailed formula, therefore the draw curve accordig to this formula used his last ame to ame ~ that is Bezier Curve. Sice most time used to draw with a computer is to operate a mouse to cotrol the path of lie segmets, which is very differet from the had-paited feelig ad the effect. Eve a shrewd paiter ca easily draw a variety of graphics, gettig the mouse to draw willfully is ot a easy matter. This is the maual work that would ever be replaced by computers, thus so far people ca oly be quite helpless. Usig the Bezier tool to draw to a great extet has made up for that deficiecy. Bezier Curve is a basic computer graphics modelig tool, ad is oe of the basic lies that is most frequetly applied i the graphical modelig. It creates ad edits graphics through cotrollig four poits i the curve (the startig poit, the edig poit ad two separate itermediate poits), i which the cotrollig lie i the ceter of the curve plays a importat role. The itermediate part of this virtual lie crosses the Bezier Curve, ad its both eds are the cotrollig edpoits. Whe move edpoits at both eds, the Bezier Curve chages the curvature (the degree of bedig); whe move the itermediate poit (that is, move the virtual cotrol lie), the Bezier Curve moves evely i the situatio of lockig the startig ad edig poits. Note that all the cotrollig poits ad odes i the Bezier curve ca be edited. Such "itelliget" vector lie provides artists with a ideal tool used for editig ad creatig graphics. This method ca easily cotrol the iput parameters (the cotrollig poits) to chage the shape of the curve, while the mathematical priciple uses the Berstei polyomial. The Bezier Curve is a curve described by usig the splie approximatio method, whose may properties make it more useful i the cofiguratio desig ad easier to implemet. Therefore, the Bezier splie has bee widely used i may graphics systems ad CAD systems. For the drawig of the Bezier Curve, ow it frequetly uses the algorithm based o the geometry, each time addig a step-legth ito a variable ad calculatig the values of other variables to calculate the poits i the curve, the coectig these poits with small straight lie segmets to geerate the curve; the algorithm requires floatig-poit operatios, ad the draw curve caot be detailed, with poor smoothess ad large amout of computatio; the other commoly used algorithm is the pixel-based algorithm, calculatig the pixel of the curve graph poit by poit with the iteger arithmetic, ad reducig the amout of computatio, able to maitai the smoothess of the curve, but as the free curve is ucertai, i.e. the tred of each sectio i the curve is irregular, if it eeds to draw the freedom curve, it requires to deped more o computers to automatically determie the directio, ad the algorithm is complex. 2. Aalyze the Bezier Curve Geeratio Algorithm 2.1 Bezier Curve Geeratio Priciple The shape of the Bezier Curve is defied through all vertices of P0, P1,..., ad Pm of a set of multilateral lie (cotrollig the polygo). Amog all vertices of a multilateral lie, oly the first poit P0 ad the last poit Pm are o the curve, while the rest poits are used to defie the order of the curve. The mathematical basis of the Bezier Curve is the Polyomial Mixed Fuctio (Harmoic Fuctio) coductig the iterpolatio betwee the first ad the last edpoits, which ca be expressed with the parametric equatio as follows: This is a -squared polyomial, with +1 items. I which, p i (i=0, 1 ) presets the vector that symbolizes the positio of the ( +1) th vertex of the polygo. p (t) = p i B i, (t) (0 t 1)

3 Guorog Xiao ad Xuemiao Xu / Physics Procedia 25 ( 2012 ) B i, (t) is the Berstei Polyomial, called as the basis fuctio, which is the harmoic fuctio presetig vectors of positios at each poit of the curve, expressed as:! B i, (t) = t i! i i (1-t) -i! I which i deotes the i th vertex, ad deotes the -squared, t as a parameter. If give +1 cotrollig poits: P k = (x k, y k, z k ), k = 0, 1, 2, the approximatig -square Bezier Curve of the characteristic polygo composed of these cotrollig poits ca be expressed as: P (u) = P kbez k, (u) 0 u 1 1 I which P k is the vector of the cotrollig poit; BEZ k, (u) is the Bezier Mixed Fuctio, usig the Berstei Polyomial to defie, i.e.: BEZ k, (u) = C (, k) u k (1-u) -k 2 I which,! C (, k) = 3 k! k! 2.2 The Bezier Curve Characteristic Aalysis x (u) = x k BEZ k, (u) y (u) = y k BEZ k, (u) z (u) = z k BEZ k, (u) Accordig to the ature of the Berstei Polyomial Berstei Basis Fuctio, it ca deduce the ature of the Bezier Curve. 1) The curve through the startig poit ad the edig poit ca prove the startig poit ad the edig poit are o the curve, providig 0 i = 0 i 0 1 i 0 Ad: 0 is 1. The developed curve is: (whe =0, 1, 2, 3 p (t) = p i B i, (t) (0 t 1) = p 0 B 0, (t) + p 1 B 1, (t)+ p 2 B 2, (t)+ p 3 B 3, (t) Whe t=0, the startig poit of the parameter ; whe i=0, the 1st vertex,! The curve p(0) = 0 0 (1-0) P 0 = P 0 1! t i =0 0 =1, The 1 st item is P 0, 0 i =0 the rest 3 items are 0 Is the startig poit P 0

4 1784 Guorog Xiao ad Xuemiao Xu / Physics Procedia 25 ( 2012 ) Whe t=1, the edig poit of the parameter ; whe i= the last vertex, The curve p(1) =!!1 1 (1-1) 0 P = P Is the edig poit P It ca be see that the curve passes the startig poit ad the edig poit of the multilateral lie. 2) The directio of taget vectors of the startig poit ad the edig poit Through solvig the derivative of the basic fuctio, it ca prove the taget vector of the vector edpoit (the startig poit ad the edig poit) is cosistet with that of the 1st ad the th (the last) edge (the same tred). The derivative of the basis fuctio:! B i, (t) = [t i! i i (1 t) -i ]!! = [i t i! i i-1 (1 t) -i ( i) t i (1 t) -i-1 ]! ( 1)! ( 1)! = [ t i-1 (1-t) -i t i (1 t) -i-1 ] ( i 1)!( i)! i!( i 1)! = [B i 1, 1 (t) B i, 1 (t)] The derivative of the Bezier Curve 1 P (t) = p i [B i 1, 1 (t) B i, 1 (t)] i, except the first item ad the last item, other items are all 0. i the startig poit t=0, i=0 P (0)= (P 1 -P 0 ) i the edig poit t=1, i= P (1)= (P -P -1 ) It ca be see that the taget vector P (0) of the startig poit is cosistet with that of the 1 st edge (P 1 -P 0 ) of the characteristic polygo. the taget vector P (1) of the edig poit is cosistet with that of the (-1) th edge (the last edge) (P - P -1 ) of the characteristic polygo. 3) The Characteristic of Covex Hull The Bezier Curve p(t) is located withi the covex hull of its cotrollig poit p i i o. The covex hull of the so-called p i i o refers to the miimum covex set cotaiig these poits. The cubic B-splie curve also possesses the covexity-preservig property, that is, if the characteristic polygo is a covex polygo, the B-splie curve is also covex istead of appearig the turig poit ad the sigular poit. 4) Curvature 2 P (t) = (p i+2 2 p i+1 + p i ) B i, 2 (t) whe t=0, p (0)= (-1)(p 2 2 p 1 +p 0 ) whe t=1, p (1) = (-1)(p 2 p =1 +p -2 ) It ca be see that the r th derivative at the edpoit of the Bezier Curve oly relates to the (r +1)th adjacet poit, othig to do with the further poits. For example: the secod derivative oly relates to

5 Guorog Xiao ad Xuemiao Xu / Physics Procedia 25 ( 2012 ) three adjacet poits, P (0) ad P0, P1, P2; while P 1 ad P-2, P-1,P or it may say that oly these poits cotribute to the curvature. 3. Bezier Curve Variable Step-Legth Algorithm 3.1 Problems Existig i the Bezier Curve Geeratio Algorithm Poits ad lies are the most basic ad commo elemets for depictig graphics; may complex graphics are all composed of poits ad lie segmets. I mathematics, a poit is a abstract positio of coordiates, o size or area. A mathematically defied lie is composed of umerous poits, oly with legth but without width. I raster graphics, a poit is expressed i pixel with a certai size ad area; while a straight lie segmet is coverted by scaig, it is determied as the best approximatio pixel sequece of the lie segmet i arrays composed of limited pixels. The maximum umber of poits that ca be displayed o the scree of a computer moitor is the pixel, determied by differet display modes of display cards, therefore, whe eter the graphical display way, firstly it eeds establishig a coordiate system o the scree of the moitor, ad the horizotal ad vertical coordiates both are take as itegers. Whe values of x ad y coordiates are ot itegers calculated by the equatio, it shall take itegers through roudig for the coordiate values. The poit-by-poit geeratio algorithm of the Bezier Curve ad the effectiveess ad rapidity showed i the curve are i direct relatioships with the selectio of the step-legth of the parameter u. I the Bezier curve geeratio algorithm with the equal step-legths, step-legths are determied based o the lie segmet with the largest curvature i the curve. If the whole curve takes this step-legth as the uified steplegth, the step-legth is reasoable for the lie segmet with larger curvature, while for the lie segmet with little chage of curvature (some places may be close to the straight lie segmet), it will cause uecessary segmetatio, thus leadig to the icrease i the umber of segmetatios. Whe the selected step-legth u is larger, the smoothess ad accuracy of the curve are relatively poor; whe the selected step-legth u is a smaller umber, the curve will be relatively smooth, but it will also occur a large umber of poits; whe display, it will be repeated due to roudig, makig the geerated efficiecy of the curve greatly reduce. Whe the selected step-legth u is larger, the draw poit of the curve is also relatively few, basically o ieffective poit, but effective poits are similarly few; such curves ofte exist defects that are rougher ad urealistic. Whe the selected step-legth u is a smaller umber, the curve will ted towards stability, ad o log icrease effective poits with the decrease of u, therefore, at the same time effective poits are tedig towards stable values, it also occur the rapidly growig pheomeo i effective poits repeatedly calculated. 3.2 The Propositio of the Variable Step-legth Curve Geeratio Algorithm I 1968, Deig.P had poited out: whe the program is implemeted, it will appear the locality regularity, that is withi a short period of time, the implemetatio of the program is oly limited to a certai part; accordigly, the memory space it accessed is also limited to a certai area. The locality priciple maily demostrates i the two aspects of time ad space limitatios. Time limitatios mea that, if a directive i the program oce has bee implemeted, it would soo be reimplemeted; if the data has bee accessed, it would still soo be accessed agai. While space limitatios mea that, the addresses accessed withi a period of time by the ruig program may focus withi a certai rage. The, accordig to the locality priciple, such problems will be also existed i the Bezier Curve Geeratio Algorithm. I the process of the curve geeratio, due to differet selectios of the step-legth of the parameter, there will be a large umber of calculatios for repeated poits, resultig i low

6 1786 Guorog Xiao ad Xuemiao Xu / Physics Procedia 25 ( 2012 ) efficiecy i the curve geeratio, if the step-legth ca be moderately varied, it ca improve the speed of the curve geeratio. 3.3 The Ideology of the Variable Step-legth Curve Geeratio Algorithm The locality priciple is the premise of virtual memory techology, which brigs elightemets to the drawig of the Bezier Curve. Whe the Bezier Curve is draw, if the selected step-legth u is smaller, ofte the smoothess ad approximatio of the curve is better, but meawhile the defect that there are more poits repeatedly calculated is also occurred. Through the aalysis of multiple test samples, it has foud that a certai regularity is existed i the times of repetitios; whe the step-legth is selected to be , 10,000 poits will be calculated, but oly 361 effective poits, while the repeated poits up to 9676; amog these repeated poits, the average repetitio each time reaches 31, ad the higher repetitio ca be up to 63 times, therefore, the itesiveess of repetitios adds the feasibility for the drawig ideology of the variable step-legth. The proposed ideology of the variable step-legth is about calculatios of the coordiates of each poit, if the rouded poit is the same as the previous poit, the step-legth u is o loger grow i fixed step, but grow i variable jumpig steps. The thought of this paper through may times of experimets has bee verified the feasibility of improvig the algorithm; at the same time of esurig the algorithm to possess higher accuracy, it reduces the times of calculatios for ieffective poits to improve the efficiecy of the curve geeratio. 4. Coclusio Aimig at defects existed i the Agle-cut Polygo Algorithm ad the Equal Step-legth Algorithm ad other Bezier Curve Geeratio Algorithms, the Variable Step-legth Algorithm proposed i this paper ot oly maitais better accuracy, but also effectively reduces a large umber of calculatios for repeated poits i the poit-by-poit geeratio algorithm. Refereces [1] Jiahua Xu, Shaqig Li, Rapid Poit-By-Poit Geeratio Algorithm Of The Ratioal Bezier Curve [J] Computer Egieerig ad Applicatios, 2004, 25: [2] Feghua Guo, Xigqiag Yag, Study O The Optimum Parameterizatio Of The Bezier Curve [J] Computers Joural, 2005: 32(5): [3] Rejiag Zhag, Guoji Wag, Improvemet O Termiatio Criteria for Ratioal Bezier Curve Dispersio [J] Software Joural, 2003:14(10): [4] Yutuo Che, Wei Big, Vector Quatizatio Of Had-Paited Carvig Patters Based o Segmeted Bezier Curve [J] Computer Egieerig, 2008,34(9):