Polyhedrons in spherical coordination system

Transcription

1 ISSN: Marcn Petrzykowsk, Int.J.Comuter Technology & Alcatons,Vol 5 (5), Polyhedrons n shercal coordnaton system Marcn PIETRZYKOWSKI Deartment of Artfcal Intellgence Methods and Aled Mathematcs, Faculty of Comuter Scence and Informaton Technology, West Pomeranan Unversty of Technology, Szczecn, 49 Żołnerska St., Szczecn, Poland E-mal: metrzykowsk@w.zut.edu.l Abstract The aer resents new way of olyhedron descrton based on shercal coordnaton system. Orgnally the method was develoed as a art of an algorthm whch wll be used to etend the method of mn-models onto the multdmensonal sace. The olyhedron s descrbed by ts faces not by ts vertces. Algorthm gves an easy-to-understand and relatvely smle way of olyhedron manulaton and testng the ont ncluson wthn the fgure area. The algorthm can be used n wde range of comutatonal geometry alcaton and s esecally handful n task n whch olyhedron sze and locaton are constantly changng. In the frst art the artcle brefly descrbes the olar and shercal coordnaton system. In the second art the aer resents olyhedrons n the shercal coordnaton system n comrehensve manner and also ts future use. Keyword: comutatonal geometry, shercal coordnaton system, olyhedron, ont ncluson 1. Introducton eometrc fgure based on shercal coordnaton system s a new dea and was develoed as a art of algorthm whch wll be used to etend the method of mn-models [1,, 3, 4, 5] onto the multdmensonal sace. Mn-model s a method of local modelng whch s learned n a heurstc way. It requres an effcent algorthm of manulaton of a olyhedron faces n the modeled sace, and algorthm testng ont ncluson wthn olyhedron area. Pont ncluson roblem s also very mortant queston whch s worked u n research [6, 7, 8]. A very mortant crteron was easness of etenson onto multdmensonal sace. The aroach descrbed n the aer can be aled to such an alcaton whch requres: check the ont ncluson wthn olyhedron area, manulaton of whole olyhedron face. Polytoes n D or 3D-sace are usually descrbed by ts vertces, laced n Cartesan coordnates system. It can be also descrbed by ts faces whch are formulated by mathematcal equatons of lnes (n Dsace) or lanes (n 3D-sace). In the second aroach, n order to fnd locaton of vertces, the comutaton of ntersectons of faces s requred. The author n the aer rooses a new knd of a olytoes descrton. The man dea s to transform all geometrc shaes nto shercal coordnaton systems 1. The dea s based on shercal coordnaton system whch s an alternatve coordnaton system commonly used n many scentfc area. Only n last few years the shercal coordnaton system was used n many felds e.g.: n hyscs for lattce Boltzmann models [9] or recse model of the terrestral gravty feld [10], n chemstry for reresentaton of roten n 3D structure [11], n medcne for facal asymmetry analyss n mandbular rognathsm atents [1], and also n mage rocessng [13, 14, 15], comuter vson [16] artcle swarm otmzaton [17], wreless networks [18] and many others.. Polar and shercal coordnaton system overvew [19, 0, 1] The olar coordnate system s two-dmensonal coordnate system n whch each ont s determned by the dstance from a center ont of the system and an angle from a fed lne. The center ont s called the ole. The dstance from a ont to the ole s called radal coordnate or smly radus. The fed lne n the system from whch angle s measured s called olar as. In the aer olar as s equvalent to a ostve art of as OX. The angle between the olar 1 D-sace s transformed nto olar coordnaton system, 3D-sace nto shercal coordnaton system. Author wll use a term shercal as more general, whch s n a fact an etenson of olar coordnaton system. localzaton of olar as has an mlcaton n transformaton formulas IJCTA Set-Oct 014 Avalable onlne@ 1658

2 ISSN: Marcn Petrzykowsk, Int.J.Comuter Technology & Alcatons,Vol 5 (5), as and a ray of the ont s called angular coordnate or olar angle. The radus s often denoted by r and the angular coordnate by, or t. Coordnates are usually lmted to r 0 and 0 < π (-π < π). The coordnates can eceed ts range. The angular coordnate lke the normal angle s maed to a value wthn the lmts 0 < π (-π < π). Negatve radus s usually nterreted as the corresondng ostve value measured n the ooste drecton. The ole can be eressed as (0; ) where s any angle, but for sake of unque reresentaton for any ont t s usually assumed that = 0. Angles n olar notaton are generally eressed n degrees or radans. A converson from Cartesan to shercal coordnates system s ossble. r y atan, y (1) where s wthn range (-π; π], atan s the common varaton on the arcus tangent functon whch gather nformaton of the sgns of the nuts n order to return the arorate quadrant of the comuted angle. y arctan y arctan y arctan undefned f 0 and y 0 f 0 and y 0 f 0 f 0 and y 0 f 0 and y 0 f 0 and y 0 () An angle n the range [0; π) may be obtaned by addng π to the value n case t s negatve: y arctan y arctan y arctan 3 undefned f 0 and y 0 f 0 and y 0 f 0 f 0 and y 0 f 0 and y 0 f 0 and y 0 (3) The reversed converson s qute smle: r cos y r sn (4) Shercal coordnaton system s an etenson of olar coordnaton system nto 3D-sace. Pont oston s determned by three numbers: the radal dstance of that ont from a fed orgn, ts olar angle measured from a fed zenth drecton, and the azmuth angle. Coordnaton of a sngle ont s a trlet (r; ; ), where r [0;1), [0; π) and = [0; π). There are many converson formulas from Cartesan to shercal coordnaton system, one of them s as follows: r arccos atan y Reverse converson s as follows: z r, y z r cos z r sn cos y r sn sn 3. eometrc fgure n shercal coordnaton system It s an assumton that the olyhedron has to be a conve set. At the begnnng we select some ont C whch s wthn olyhedron area. The ont C s the center of the shercal coordnaton system and all other onts are transformed to shercal coordnaton. Each olyhedron face F s transformed n the followng way. At the begnnng we fnd an equaton : + y + z = of a lane whch contans the face. Then we fnd a ont whch s a roecton of ont C on the lane. Pont s called face generaton ont. Knowng the equaton of the lane, the vector v,, whch s orthogonal to the lane s also known. Resolvng smle equaton (7) Cartesan coordnates of the ont s founded: C t y yc t z zc t y z (7) (5) (6) IJCTA Set-Oct 014 Avalable onlne@ 1659

3 ISSN: Marcn Petrzykowsk, Int.J.Comuter Technology & Alcatons,Vol 5 (5), where t R. For further comutaton for any ont we have to defne vector = { }. It contans the angle values between vector created by ont and vectors created by generaton onts 1 ;... ; J. The angles value can be comuted usng dot roduct, for Cartesan coordnates: arccos y y z y z y z z (8) and for shercal coordnates (after smlfcaton) we have: arccos cos cos sn sn cos (9) Each olyhedron contans J faces (e.g.: for cube J = 6, for smle J = 4). The lane whch contans olyhedron face wll be called smly face and s defned as a grou of onts: r F, : r (10) cos The face n fact dvdes all the sace nto two halfsaces. The frst half-sace conssts of data onts whch may be ncluded nto the olyhedron area. The set of onts whch may be ncluded by face s defned as: r I : r cos (11) 3.1. Polygons n olar coordnaton system Polygons descrton n D-sace s smlar to olyhedrons n 3D-sace. At the begnnng we should defne a ont C whch becomes the ole. Then we fnd the equaton l: + y of a lne whch contans the fgure face. Then we fnd a ont by resolvng smle equaton: C t y yc t y (14) where t R. We also defne the vector = { } where for Cartesan coordnates we have: arccos y y y y (15) and for shercal coordnates after smlfcatons we obtan: (16) Other formulas (10-13) are dentcal as n 3D-sace. The stuaton s descrbed on Fg. 1. Data onts marked by trangles are certanly ecluded from olygon (trangle) area whlst onts marked by squares are ossbly ncluded nto the fgure by currently selected face. But only few data onts wll be ncluded, some of them wll be ecluded by another faces. The second half-sace conssts of data onts whch are certanly ecluded from the fgure area. The set of onts certanly ecluded by face s defned as: E : r r cos (1) Every face dvdes the sace n such way. Intersecton of half-saces (whch nclude onts) of all faces shows whch onts are ncluded nto the olyhedron area. Set of onts Z ncluded n the olyhedron we denoted as: Z I I 1 I J (13) IJCTA Set-Oct 014 Avalable onlne@ 1660

4 ISSN: Marcn Petrzykowsk, Int.J.Comuter Technology & Alcatons,Vol 5 (5), can be comuted by (10) and by smle trgonometrc calculaton based on cosne. The most mortant advantage s that the algorthm could be very easy etended to hgher dmensonal sace. In the general the olyhedron n shercal coordnaton system can be aled to a wde range of alcaton whch requres: test the ont ncluson wthn fgure area and manulaton of whole fgure face. Algorthm s esecally handful n task n whch olyhedron sze and locaton are constantly changng. It should be noted that change of varous arameters of the face (face generaton ont) has dfferent mlcatons. If we change the radus r of the face F (change the dstance between face and center of the coordnaton system) to check the ont ncluson we only check followng condton: r r (17) cos Fg. 1. Eamle of trangle based on olar coordnaton system n D-sace. 4. Alcatons Polytoes based on shercal coordnaton system were develoed as a art of an algorthm of mn-models. The concet of mn-models method was develoed by Pegat [1, ]. Mn-model oerates on a data n the local neghborhood of the query ont that actually s mortant for a scentst. The method requres relatvely small number of data samles n the learnng rocess. Algorthms descrbed n ths aer rovde a relatvely smle and easy-to-understand way of olyhedrons manulaton, whch s deal n the task of creatng local neghborhood of the query ont (query ont neghborhood s equvalent to fgure area). It also gves a convenent way of check the ont ncluson wthn fgure area. Algorthms descrbed above can be use n every task nvolvng fgure manulaton. Ths aroach frees us from the course of dmensonalty because e.g.: hyercube n the general number of dmenson has n vertces but only n faces. It has other very mortant advantages. Manulaton of a face s relatvely smle and s reduced to manulaton of three arameters whch are equvalent to the face generaton ont coordnates (r; ; ). In descrton of olytoes based on ts vertces, manulaton of one face requres translaton of all vertces, keeng n mnd the fact they have to be colanar. Thus, manulaton of a sngle verte locaton n a sace grater then - dmensonal s mossble. Manulaton by faces frees us from that roblem. Incluson of ont wthn fgure If the angular coordnatons of the face were changed we have to comute new value of the angle and we use (10). But t should be noted that f only one angle was changed some art of the equaton can be stored n the memory (art whch contans value of unchanged angle) n order to mnmze comutatonal effort. 5. Concluson In the aer new way of olyhedron descrton based on shercal coordnaton system was resented. The algorthm s characterzed by relatvely smle and easy-to-understand way of fgure manulaton, and gves convenent way of checkng the ont ncluson wthn olyhedron area. All mathematcal oeratons used for algorthms were descrbed n comrehensvely manner. Author n future research wll move towards olytoes based on n-shercal coordnaton system n general number of dmensons. The future work wll also contan the use of the algorthm n the mn-model alcaton. 6. References [1] A. Pegat, B. Waskowska and M. Korzeń, Alcaton of the self-learnng, 3-ont mn-model for modellng of unemloyment rate n Poland [n Polsh] Studa Informatca, nr 7, Unversty of Szczecn, 010, , 010. [] A. Pegat, B. Waskowska and M. Korzeń, Dfferences between the method of mn-models and the k-nearest neghbors an eamle of modelng unemloyment rate n Poland, Informaton Systems n management IX-Busness Intellgence and Knowledge Management. WULS Press, Warsaw, 011, [3] M. Petrzykowsk, Comarson of effectveness of lnear mn-models wth some methods of modellng, Młodz Naukowcy dla Polske Nauk, Kraków, 011, IJCTA Set-Oct 014 Avalable onlne@ 1661

5 ISSN: Marcn Petrzykowsk, Int.J.Comuter Technology & Alcatons,Vol 5 (5), [4] M. Petrzykowsk, The use of lnear and nonlnear mnmodels n rocess of data modellng n a D-sace. Nowe Trendy w Naukach Inżynerynych, Kraków, 011, [5] M. Petrzykowsk, Effectveness of mn-models method when data modellng wthn a D-sace n an nformaton defcency stuaton, Journal of Theoretcal and Aled Comuter Scence, 6(3), 01, [6] W. L, E. Ong, S. Xu and T. Hung, A ont ncluson test algorthm for smle olygons, In: COMPUTATIONAL SCIENCE AND ITS APPLICATIONS - ICCSA 005, PT 1, volume 3480 of LECTURE NOTES IN COMPUTER SCIENCE, SPRINER-VERLA BERLIN, 005, , ISBN , ISSN [7] R. Soukal, I. Kolngerova, Star-shaed olyhedron ont locaton wth orthogonal walk algorthm, In: ICCS INTERNATIONAL CONFERENCE ON COMPUTATIONAL SCIENCE, PROCEEDINS, volume 1 of Proceda Comuter Scence, ELSEVIER SCIENCE BV, 010,. 19 8, ISSN [8] H. Wu, J. ong, D. L and W. Sh, An algebrac algorthm for ont ncluson query, COMPUTERS & RAPHICS-UK, 4(4), 000, , ISSN [9] V. E. Ambrus and V. Sofonea, Hgh-order thermal lattce Boltzmann models derved by means of auss quadrature n the shercal coordnate system, PHYSICAL REVIEW, 86(1, ), 01, ISSN [10] R. van der Toorn and J.T.F. Zmmerman, On the shercal aromaton of the geootental n geohyscal flud dynamcs and the use of a shercal coordnate system, EOPHYSICAL AND ASTROPHYSICAL FLUID DYNAMICS, 10(4), 008, , ISSN [11] V. M. Reyes, Reresentaton of Proten 3D Structures n Shercal (rho, h, theta) Coordnates and Two of Its Potental Alcatons, INTERDISCIPLINARY SCIENCES- COMPUTATIONAL LIFE SCIENCES, 3(3), 011, , ISSN [1] S. J. Yoon, R. F. Wang, H. S. Hwang, B. C. Kang, J. S. Lee and J. M. Palomo, Alcaton of shercal coordnate system to facal asymmetry analyss n mandbular rognathsm atents, Imagng Scence In Dentstry, 41(3), 011. [13] W. S. Lao, T. J. Hseh, Y. L. Chang, PU Parallel Comutng of Shercal Panorama Vdeo Sttchng. In: PROCEEDINS OF THE 01 IEEE 18TH INTERNATIONAL CONFERENCE ON PARALLEL AND DISTRIBUTED SYSTEMS (ICPADS 01), IEEE, 01, , ISBN , ISSN [14] R. Moreno, M. rana, D. M. Ramk and K. Madan, Image segmentaton on shercal coordnate reresentaton of RB colour sace, IET IMAE PROCESSIN, 6(9), 01, , ISSN [15] D. Zhang, X. Kang, J. Wang, A novel mage de-nosng method based on shercal coordnates system, EURASIP JOURNAL ON ADVANCES IN SINAL PROCESSIN, 01, ISSN [16] W. Zhang, J. Me, Y. Dng, T. Huang, Angle recognton algorthm based on dscrete seres data of obectboundary n olar coordnate, OPTIK, 14(1), 013, , ISSN [17] I. Cruz-Aceves, J.. Avna-Cervantes, J. M. Loez- Hernandez and S. E. onzalez-reyna, Multle actve contours drven by artcle swarm otmzaton for cardac medcal mage segmentaton, Comutatonal And Mathematcal Methods In Medcne, 013. [18] S. Tao, A. L. Ananda, M. C. Chan, Shercal Coordnate Routng for 3D Wreless Ad-hoc and Sensor Networks. In: 008 IEEE 33RD CONFERENCE ON LOCAL COMPUTER NETWORKS, VOLS 1 AND, volume 008, IEEE Com Soc, IEEE, 008, , ISBN ISSN [19] I. Bronshten, K. Semendyayev,. Musol and H. Muhlg, Handbook of Mathematcs, Srnger, 007. ISBN [0] A. Polyann and A. Manzhrov, Handbook of Mathematcs for Engneers and Scentsts, Taylor & Francs, 010. ISBN [1] P. Moon and D. Sencer, Feld theory handbook: ncludng coordnate systems, dfferental equatons, and ther solutons, Srnger-Verlag, ISBN IJCTA Set-Oct 014 Avalable onlne@ 166