, 49-55 Receved: 2014-11-12 Accepted: 2015-02-06 Ole publshed: 2015-11-16 DOI: http://dx.do.org/10.15414/meraa.2015.01.02.49-55 Orgal paper Bezer curve ad ts applcato Duša Páleš, Jozef Rédl Slovak Uversty of Agrculture, Faculty of Egeerg, Departmet of Mache Desg, Ntra, Slovak Republc ABSTRACT Descrpto of the Bezer curve s preseted. We expla detal creato of the calculato algorthm together wth the resultg program. It also cludes drawg of the base fuctos of the Berste polyomals. Frstly, the procedure s appled to the theoretcal example gve by te cotrol pots a plae whch approxmate the Bezer curve. Secodly, the applcato whch we have gve 138 pots of trajectory of real vehcle. Pots are located space ad we use them aga for approxmato of the smooth Bezer curve. KEYWORDS: Bezer curve, Berste polyomal, curve fttg. JEL CLASSIFICATION: M55, N55 INTRODUCTION The curves ca be determed usg cotrol pots, to whch are usually added eve further restrctos, such as boudary codtos. The cotrol pots are used ether to terpolate the curve, whe costructed smooth curve pass through all the gve pots, or to approxmate curve whe smooth curve pass oly some selected cotrol pots or goes off these pots 1. Typcal examples are the Lagrage terpolato, Hermte terpolato or Newto terpolato. The best kow approxmato method s the approxmato method of the least squares. I ths artcle we preset approxmato method usg Bezer curve. Correspodg author: Duša Páleš, Slovak Uversty of Agrculture, Faculty of Egeerg, Departmet of Mache Desg, Tr. A. Hlku 2, 949 76 Ntra, Slovak Republc. E-mal: dusa.pales@uag.sk
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 MATERIAL AND METHODS Bezer curve Its ame was gve by Frech egeer Perre Bezer, who worked at the Frech car factory Reault. A smple algorthm for creatg Bezer curve costructed compettor's employee of Ctroe Paul de Casteljau. Both desgers have publshed ther results the sxtes of the prevous cetury 6. De Casteljau algorthm s based o the repeated use of lear terpolato ad geeralze the costructo of parabolc curves for hgher orders. Fg. 1 Illustrato of the de Casteljau algorthm for four-pot cotrol polyomal P 0 2,3, P 1 4,21, P 2 15,28, P 3 28,19 Polygo was specfed by four pots the plae as show the Fgure 1 3. Smlarly, we could have determed cotrol pots space ad the process would work aalogously. Bezer curve was expressed parametrcally, the parameter t 0, 1. Pots of cotrol polygo we deoted as 0-th approxmato pot of the curve (subscrpt represets the seral umber of pot ad superscrpt approxmato order) P,P,P,P P,P,P,P (1) 0 0 0 0 0 1 2 3 0 1 2 3 The frst approxmato s obtaed from the zero approxmato usg relatos P (t) (1 t).p t.p 1 0 0 0 0 1 P (t) (1 t).p t.p 1 0 0 1 1 2 P (t) (1 t).p t.p P,P,P 1 0 0 1 1 1 2 2 3 0 1 2 (2) Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 50
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 Ad aalogous cotue the secod ad thrd approxmato P (t) (1 t).p t.p 2 1 1 0 0 1 P (t) (1 t).p t.p P,P 2 1 1 2 2 1 1 2 0 1 3 2 2 3 0 0 1 0 P(t) P (t) (1 t).p t.p P (4) The pot of the thrd approxmato P 3 0 s the pot of the curve for etered parameter value t. Ths procedure should be repeated for each value t 0, 1. For t 0, P(0) P0 ad for t 1, P(1) P3, therefore Bezer curve always passes through the frst ad the last pot. If to the relato (4) successvely substtute relatos (3), (2) ad (1) we obta the parametrc represetato of the curve the form (3) P(t) (1 t).p 3.(1 t).t.p (1 t).t.p t.p 0 3 2 2 3 0 1 2 3 3 3. 1 t. t.p (5) The parameter t appears at most the cube, so t s a cubc Bezer curve. Repeat the procedure creates a tragular approxmato scheme of successve pots P P P P P P P P 0 0 0 0 0 0 1 1 2 2 3 3 P P P 1 1 1 0 1 2 P 2 2 0 1 P 3 0 P P(t) (6) RESULTS AND DISCUSSION Numercal example for may cotrol pots The process ca be geeralzed to ay umber of cotrol pots of the polygo 4. Whe the polygo has 1 pots t s ecessary to perform steps to get the pot of the curve. Bezer curve of degree the parametrc represetato has the form 1 p( t) 1 t t P1 (7) 1 1 Taget of the curve obtaed at pot P(t) s drectly determed by pots P 0-1, P 1-1, the case of our four-pot polygo (6) hece by pots P 0 2, P 1 2. The taget at the startg pot of the curve s the same as the frst cotrol edge of the polygo ad lke maer taget the last pot of the curve s detcal to the last cotrol edge of the polygo. Formula 1 p( t) 1 t t P1 (8) 1 1 s called the Berste polyomal of degree. Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 51
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 Wth t looks smply parametrc wrtg of Bezer curve P t) B ( t) P. (9) ( 1 Berste polyomals form a bass of the vector space for polygo degree at most. They are umodal (dromedary) fuctos wth a sgle maxmum at the pot t. I Fgure 2 we preset the Berste polyomals for 9. Fg. 2 Berste polyomals for =9 For 9 we chose accordg to 2 te pots the plae ad developed a algorthm (10) to calculate the Bezer curve wth step parameter t 0. 05. The umber of pots the plae as well as the step parameter ca vary. For plaar curve we use a algorthm twce both drectos x ad y. The selected cotrol polygo ad the resultg curve s show Fgure 3. Bezer _ Curve(, tt, x): k 0 for t 0, tt...1 BC! BC k. 1 t.t.x k k 1 (10) 0!.! Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 52
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 Applcato of Bezér curve Fg. 3 Bezer curve draw for cotrol pots P 0 14,0, P 1 6,4, P 2-4,16, P 3 4,19, P 4 14,30, P 5 16,30, P 6 26,19, P 7 5,16, P 8 24,4, P 9 16,0, the parameter value tt=0.05 The algorthm was used also three dmesoal space by applcato (10) the drectos x, y ad z 5. We solved the real movemet of the vehcle ad record ts posto through 138 dscrete pots. These have served as cotrol pots for the Bezer curve. I Fgure 4 we preset a polygo, whch was created by coectg pots wth le segmets ad Fgure 5 shows the Bezer curve, whch smoothed sequece of movemet of the vehcle. Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 53
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 Fg. 4 Orgal trajectory of vehcle from cotrol pots coected wth les CONCLUSIONS Fg. 5 Approxmated trajectory of vehcle by Bezér curve The resultg regstrato of the algorthm (10) proved to be relatvely smple, regardless of the umber of cotrol pots ad also of the parameter sze tt of Bezer curve. Curve fttg ca be doe a plae ad space as show the preseted examples. Practcal applcato of the real movemet of the vehcle space replaced polygo composed from le segmets by smooth Bezer curve, for whch there are ot dscotuous pots of the frst dervato. Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 54
Mathematcs Educato, Research ad Applcatos (MERAA), ISSN 2453-6881 ACKNOWLEDGEMENT The research performed at the Departmet of Mache Desg of the Faculty of Egeerg of the Slovak Uversty of Agrculture Ntra was supported by the Slovak Grat Agecy for Scece uder grat VEGA No. 1/0575/14 ttled Mmzg the rsks of evrometal factors amal producto buldgs. REFERENCES [1] Bastl, B. Bezérove krvky. Plzeň: Západočeská uverzta v Plz. [ct. 2014-12-18]. Retreved from http://geometre.kma.zcu.cz/dex.php/www/cotet/dowload/1152/3264/ fle/gpm_bezer.pdf?phpsessid=05c59e8bf9cc6c31b703704908efa66e [2] Macková, B. ad Zaťková, V. (1985). Rešee základých úloh z deskrptívej geometre pomocou počítača. Bratslava: SVŠT Bratslava. [3] Sederberg, T. W. (2012). Computer Aded Geometrc Desg Course Notes. Brgham Youg Uversty, 262 p. [ct. 2014-12-18]. Retreved from http://cagd.cs.byu.edu/~557/text/cagd.pdf [4] Shee, C. K. (2014). Itroducto to Computg wth Geometrc Notes. Departmet of Computer Scece, Mchga Techologcal Uversty. [ct. 2014-12-18]. Retreved from http://www.cs.mtu.edu/~shee/courses/cs3621/notes/ [5] Rédl J., Páleš D., Maga J., Kalácska G., Válková V., Atl J. (2014). Techcal Curve Approxmato. Mechacal egeerg letters. Szet Istvá Uversty, Gödöllö. Revewed by 1 Mlada Balková, Slovak Uversty of Agrculture, Faculty of Egeerg, Departmet of Buldgs, Tr. A. Hlku 2, 949 76 Ntra, Slovak republc 2 Igrd Karadušovská, Slovak Uversty of Agrculture, Faculty of Egeerg, Departmet of Buldgs, Tr. A. Hlku 2, 949 76 Ntra, Slovak republc Slovak Uversty of Agrculture Ntra :: Departmet of Mathematcs, Faculty of Ecoomcs ad Maagemet :: 2015 55