VISUALIZATION OF FREE FORM SURFACES BY A RAY TRACING METHOD BRĪVAS FORMAS VIRSMU VIZUALIZĀCIJA AR STARU TRASĒŠANAS METODI

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1 COMPUTER SCIENCE ISSN DATORINĀTNE 7-3 TECHNOLOGIES OF COMPUTER CONTROL DATORVADĪAS TEHNOLOĢIJAS VISUALIATION OF FREE FORM SURFACES A RA TRACING METHOD RĪVAS FORMAS VIRSMU VIUALIĀCIJA AR STARU TRASĒŠANAS METODI Aleksandr Ssoe PhD stdent Rga Techncal Unersty Faclty of Compter Scence and Informaton Technology Insttte of compter control atomaton and compter engneerng Address: Meza Str. 1/3 LV-148 Rga Lata E-mal: alex@nbox.l Aleksandrs Glazs Dr.habl.sc.ng. professor Faclty of Compter Scence and Informaton Technology Insttte of compter control atomaton and compter engneerng Address: Meza Str. 1/3 LV-148 Rga Lata E-mal: glaz@egle.cs.rt.l. Keyords: Ray tracng free-form srfaces ézer srfaces 1. Introdcton Problem of salzaton of free form srfaces s actal n aros areas of a scence and engneerng. One of mathematcal models sed for ths prpose s the mathematcal descrpton of a ézer srface [1]. Classcal methods the compter graphcs based on polygonal models [] se only polygonal nterpolaton of a srface [1]. It netably condcts to occrrence of an error and as a conseqence to dscrepancy of an mage. Modern tools of compter graphcs sch as graphc lbrares OpenGL Drect ge opportntes for salzaton of ézer srfaces bt they hae all classcal methods dsadantages. Especally t s shon n sal effects. For example n shadng reflecton and procedral textrng. Alternate of methods sold n OpenGL s the ray tracng. Sch approach has a fe adantages: allos to render exact model of a free form srface ézer allos th hgh accracy to realze sal effects hch renderng n classcal methods s nconenent. A knon approach that ses ray tracng s descrbed n ork [4]. t that ork sggests sng mnmzng areas to fnd the ht pont of ray and srface. Whle the approach descrbed n gen ork s based on sng random search and gradent 7

2 method to fnd the ht pont of a ray and srface. The gen ork s deoted to se of ths approach for salzaton of ézer srfaces on an example of a bqadratc srface.. Algorthm of a ezer srface salzaton Let's consder algorthm of salzaton of a bqadrc srface hch descrbed by ézer srface. As knon [1] n ths case bqadrc srface descrbed as a ézer srface: P here: P control ponts the ernsten polynomals of the nd poer parameters. y analogy a ray can be represented as: 1 1 t P t 1 here: P control ponts 1 t the ernsten polynomals of the 1st poer; t parameter. The mathematcal task of a ray and a srface ht pont search can be descrbed as the nonlnear eqatons system: t t 3 t To defne nknon parameters and t of the aboe eqaton system shold be consdered a case hen the ray concdes th one of coordnate axes Oz. Then the system of the eqatons s defned as follos: 4 t The solton of the system of the eqatons [4] can be obtaned by solng an optmzaton task. As for fndng parameters and only frst eqatons to be consdered therefore the fncton of mnmzaton can be descrbed as follos: + mn 5 Task of fncton optmzaton shold be dded nto to parts: prelmnary search and optmzaton of the parameters specfcaton of roots..1. Prelmnary search The prelmnary search s based on a method of Monte-Carlo. Frst of all t s necessary to generate set M of random ponts ; [1;N]. From ths set e choose 8

3 9 a sbset of ponts M for the sbseqent optmzaton.e. ponts ; M hch can be defned accordng to the follong neqalty: [ ] N * * ε 6 here: coordnates of a -th pont * * coordnates of a pont that can be fond by follong mnmzaton fncton: mn * * ε the gen accracy N nmber of random ponts. The frther optmzaton of fncton s carred ot based on these sbset ponts.. Optmzaton of parameters For optmzaton e choose a gradent method [3]. The ales for these parameters are defned at each search step as follos: h h old ne old ne 7 here: P P P P 8 h eght orkng step. The optmzaton s carred ot ntl the condton ll be exected: ε 9 here ε small nmber n ths paper e assme ε<<1. After optmzaton t s necessary to choose from the trned ot sbset only one pont hch ll be the decson of system of the eqatons 3 for parameters and. As the decson parameters and hae be chosen so that the ale of fncton ll be mnmal: mn 1

4 After a fndng of ales of parameters and thrd parameter t can be obtaned from 3rd eqatons of system 3 as follos: here: P t 11 P P1 coordnates of srface P P 1 coordnates of control ponts of a ray. For a fndng of a ht pont of a srface th any ray t s sffcent th the help of elementary geometrcal transformatons to transforme sch a ray and control ponts of a srface the ay that the ray old concde th a necessary coordnate axs n or example - axs. 3. Expermental reslts In ths secton the practcal reslts of applcaton of the consdered approaches are presented. Fg.1 presents the mages of a bqadrc srface thot modelng of llmnaton receed by ray tracng and for comparson - classcal methods realzed n lbrary OpenGL. A comparson of Fg.1-a th Fg.1-b demonstrates that at constrcton of srface the ray tracng does not hae any adantages n comparson to OpenGL lbrary. In Fg. the mages bqadrc of a srface th modelng of llmnaton receed ray tracng and for comparson - th applcaton OpenGL are gen. The comparson of Fg.-a th Fg.-b demonstrates that at modelng of llmnaton the ray tracng has adantage compared to OpenGL lbrary. In Fg.3 the scene th sal effects llmnaton shadng raster textrng procedral textrng receed by ray tracng s gen. For comparson - classcal methods realzed n lbrary OpenGL are gen. The comparson of Fg.3-a th Fg.3-b demonstrates that at modelng of sal effect the ray tracng assres some adantages to compare th OpenGL In Fg.4 the scene th realzaton of sch sal effects as s gen: llmnaton shadng reflecton raster textrng procedral textrng. Consderng that the standard tools for modelng of reflecton n lbrary OpenGL are not presented the comparsons n ths problem are mpossble. 1

5 Fgre -a: A bqadrc srface th llmnaton Ray Tracng. Fgre -b: A bqadrc srface th llmnaton OpenGL. 11

6 Fgre 3-a: Scene th textrng and shadng Ray Tracng. Fgre 4-a: Scene th bqadrc srfaces Ray Tracng. Fgre 4-b: Scene th bqadrc srfaces OpenGL. Fgre 5-a: Scene th bcbc srfaces Rat tracng Fgre 5-b: Scene th bcbc srfaces OpenGL. 1

7 Fgre 6: Scene th textrng shadng and reflecton Ray Tracng 4. Conclson The receed reslts reflected n a fg. 1-4 sho that: Ray tracng does not ge adantages at constrcton of srfaces at absence of llmnaton. The classcal methods th qte precse nterpolaton cope th ths task. Ray tracng ges adantages at constrcton of srfaces th modelng of llmnaton. The classcal methods sole ths task th orse qalty. Ray tracng allos correctly and precsely flfll modelng of shadng. The methods hch are realzed n OpenGL do not allo ths and reqre deelopng of addtonal algorthms for ths task Ray tracng allos correctly and precsely execte modelng reflecton ncldng from free form srfaces. The methods realzed n OpenGL do not allo t and the deelopment of separate algorthms for realzaton of ths task reqred. 5. Acknoledgements Ths ork has been partly spported by the Eropean Socal Fnd thn the Natonal Programme Spport for the carryng ot doctoral stdy programm s and postdoctoral research proect Spport for the deelopment of doctoral stdes at Rga Techncal Unersty 6. References 1. D.F. Rogers J.A. Adams. Mathematcal Elements for Compter Graphcs nd Ed. McGra-Hll oston MA D. Hearn M.P. aker Compter Graphcs th OpenGL 3 rd Ed. Prentce Hall 4. 13

8 3. Д. Химмельблау Прикладное нелинейное программирование. М.: Мир T. Nshta T.W. Sederberg and M. Kakmoto. Ray Tracng Trmmed Ratonal Srface Patches. Compter Graphcs Ssoes A. Glazs A. rīas formas rsm zalzāca ar star trasēšanas metod Darbā tek apskatītas blneāro rsm zalzācas metode n to agstas kaltātes attēl egūšana. To darba metodes balstās z star trasēšanas model. lneāra rsma n stars tek aprakstīt parametrskā edā krš balstās z ernštena bāz. Apskatītas tās sastādaļas īpašības kras rodas star trasēšanas zmantošanas gadīmā. Sysoe A. Glaz A. Vsalzaton of free form srfaces by a ray tracng method In the present ork the method of renderng of free form srfaces and creatons of ther hghprecson mages s consdered. Ths method s based on a model of ray tracng. A free - form srface and ray are descrbed n a parametrcal knd hch s based to base of ernshten. Those featres component are consdered hch occr n case of se of ray tracng. Сысоев А. Глаз А. Визуализация криволинейных поверхностей методом трассировки лучей В работе рассмотрен метод визуализации произвольных криволинейных поверхностей и создания их высокоточных изображений. Этот метод основывается на модели трассировки лучей. Произвольная криволинейная поверхность и луч описываются в параметрическом виде на основе базиса Бернштейна. Рассмотрены те особенности составляющих которые появляются в случае использования трассировки лучей. 14