Research Article A Comparison of Standard One-Step DDA Circular Interpolators with a New Cheap Two-Step Algorithm

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1 Modelling and Siulation in Engineering, Article ID , 6 pages Research Article A Coparison of Standard One-Step DDA Circular Interpolators with a New Cheap Two-Step Algorith Leonid Moroz, 1 Jan L. CieVliNski, 2 Marta Stakhiv, 1 and Volodyyr Maksyovych 1 1 Lviv Polytechnic National University, S. Bandery Street 12, Lviv 79013, Ukraine 2 Uniwersytet w Białystoku, Wydział Fizyki, ul. Lipowa 41, Białystok, Poland Correspondence should be addressed to Jan L. Cieśliński; janek@alpha.uwb.edu.pl Received 28 June 2013; Accepted 21 October 2013; Published 20 January 2014 Acadeic Editor: Jing-song Hong Copyright 2014 Leonid Moroz et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We present and study existing digital differential analyzer (DDA) algoriths for circle generation, including an iproved two-step DDA algorith which can be ipleented solely in ters of eleentary shifts, addition, and subtraction. 1. Introduction Digital interpolation algoriths are widely used in achine tools with nuerical control, graphics displays and plotters, and anipulation robots. Circles and circular arcs frequently appear in coputer graphics, coputer-controlled printing, and autoated control; see [1]. One of ost popular ethods for generation of circles and arcs is known as digital differential analyzer (DDA). Digital circular interpolators (or angular sweep generators) based on DDA approach are widely used [2 6]. There are any papers where characteristics research results of such interpolators are systeatized [7, 8]. In this paper a new iproved two-step algorith for DDA circle generation is presented. In paper [9] the idea of applying a Nystro two-step schee to circle generation appeared for the first tie. Our next paper [10] developed theoretical and geoetric aspects of this ethod. In the present paper we focus on practical and experiental issues. Moreover, here we consider circles of arbitrary radius R, while in [9, 10] we assued R=1. The accuracy of this ethod is higher than the accuracy of other known algoriths. Because of its siplicity (it uses only eleentary shift, addition, and subtraction); this ethod canalsobeusedinnuericalcontrol,planningechaniss, and so forth. 2. DDA Algoriths The general class of DDA algoriths for circles generation is based on obvious trigonoetric transforations describing rotation of vector R in the coordinate plane x-y (Figure 1); x n+1 =Rsin (α+δα) =Rsin α cos Δα + R cos α sin Δα =x n cos Δα + y n sin Δα, y n+1 =Rcos (α+δα) =Rcos α cos Δα + R sin α sin Δα =y n cos Δα x n sin Δα. Advantages of DDA algoriths include siplicity and high speed in generating circle point coordinate x n+1, y n+1. Substituting Δα=ε=2,where is an integer (usually 3), we rewrite (1)as x n+1 =x n cos ε+y n, (1) y n+1 =y n cos ε x n, (2) which, obviously, can be expressed by a rotation atrix A: cos ε A=[ ]. (3) cos ε

2 2 Modelling and Siulation in Engineering y Another second order schee, different for truncated Taylor series, is proposed in [2]: α Δα x n, y n x n+1, y n+1 R x 2 )x n +(ε ε3 4 )y n, y n+1 =(1 ε2 2 )y n (ε ε3 4 )x n, det A=1 ε4 4 + ε6 16. All the above DDA algoriths have systeatic radial error at each step, which is defined as (7) Figure 1: Rotation of vector R in the coordinate plane x-y. To avoid expensive coputation of trigonoetric functions DDA algoriths use sipler (cheaper) expressions instead of cos ε and. For exaple, by replacing trigonoetric functions by truncated Taylor series one gets socalled siultaneousdda algoriths [3]. The deterinant of the rotation atrix is 1. Matrices of DDA algoriths have different deterinants and det A 1 only approxiately. The closer it equals 1 the ore accurate is the corresponding circular interpolator [2]. Approxiating cos ε 1and ε, we obtain the siplest (and least accurate) DDA algorith [2, 11]: x n+1 =x n +εy n, y n+1 =y n εx n, det A=1+ε 2. (4) Much ore accurate algorith is obtained using cos ε 1 (ε 2 /2) and ε[7, 12, 13]: 2 )x n +εy n, y n+1 =(1 ε2 2 )y n εx n, ΔR n = x 2 n +y2 n R. (8) In [7], given the recurrence relations for algoriths that do not have radial errors x n+1 = 4 ε2 4+ε 2 x n + 4ε 4+ε 2 y n, y n+1 = 4 ε2 4+ε 2 y n 4ε 4+ε 2 x n. However, ipleentation of these algoriths requires floating pointoperations [7]. Replacing the coefficients of (9)with the first two ters of Taylor series [2], we obtain (9) 4 ε 2 4+ε 2 1 ε2 2, 4ε 4+ε 2 ε ε3 4. (10) As a result we obtain, obviously, algorith given by (7). A lower level of radial error with uch sipler hardware ipleentation can be achieved using the following algorith [3, 13]: 2 )x n +(ε ε3 8 )y n, det A=1+ ε4 4. (5) y n+1 =(1 ε2 2 )y n (ε ε3 8 )x n, (11) Approxiating cos ε and with first two ters of Taylor series, we obtain the following algorith [14]: 2 )x n +(ε ε3 6 )y n, y n+1 =(1 ε2 2 )y n (ε ε3 6 )x n, det A=1 ε4 4 + ε6 36. (6) det A=1+ ε6 64. Another schee worthwhile to be entioned is the so-called agic circle interpolator [5, 7, 15]: x n+1 =x n +εy n, y n+1 =(1 ε 2 )y n εx n, det A=1. (12) This interpolator is ore accurate than the siplest schee (4) and requires the use of only two ultipliers. However, in practice it is not used for large ε, because then the generated circle becoes an ellipse [2, 15].

3 Modelling and Siulation in Engineering 3 Table 1: Maxiu values of the absolute radial error (R =2 ). Nuber of algoriths =3 =5 =6 =8 =10 =12 (4) (5) e e e e e 7 (6) 0.41e e e e e e 7 (7) e e e e e 7 (11) 0.12e e e e e e 15 (12) (13) (14) e e e e 3 (15) (16) 0.18e e e e e e 2 There is also a group of sequential DDA algoriths [12]. Here are soe of the: x n+1 =x n +εy n, y n+1 =y n εx n+1, (13) x n+1 =x n +εy n, y n+1 =y n εx n+1, y n+2 =y n+1 εx n+1, x n+2 =x n+1 +εy n+2, (14) 2 )x n +εy n, y n+1 =(1 ε2 2 )y n εx n+1. Two-step algorith: (15) x n+2 =x n+1 cos ε x n, y n+2 =y n+1 cos ε y n (16) with initial conditions: x 0 = R, y 0 = 0, x 1 = Rcos ε, and y 1 = R, isknownasa direct orbiquadforofthe circular interpolator [14]. Note that its ipleentation needs only two ultipliers. In the process of circle points generation radial errors accuulate, and the law of accuulation is specific to a given algorith. Note that in ost cases, the circle radius is chosen to satisfy condition 2 1 R 2 [2, 12, 15]. Obviously, the greatest error will occur when R=2.InTable 1 we give axiu values of the absolute radial systeatic errors for R=2 and n=2π(1/ε)=[2π2 ], that is, for the full circle generation. Initial conditions are chosen to be x 0 =0, y 0 =R. Apart fro radial error an iportant role in such devices (especially in coputer nuerical control (CNC) interpolators) is played by the chord error E ch [7], which should not exceed one basic length unit (BLU). Unlike radial error, the chord error does not accuulate. In [7] itisshownthatfor faster circle point generation for ost accurate ethods (5) (7), (9) Δα canbechosenfroconditionδα = 8/R at E ch =1(we recall that Δα ε); that is R E ch or R (17) Since the CNC systes radius can reach value R ax =10 8 BLU, the CNC interpolators angles range Δα = ε = 2, where =3, 12 overlapping all possible radius values range (the though adopted algoriths can work and at >12). In Table 2 wegiveaxiuvaluesofabsoluteradial systeatic error for ost accurate algoriths in case R = Fro these results we see that algorith (11) is the ost accurate. Therefore, in next sections we choose algorith (11) in order to copare it with two new circular interpolators introduced in [10]. 3. New DDA Algoriths In the proposed two-step DDA algorith the values x n+2 and y n+2 are deterined by the following forula: x n+2 =x n 2 y n+1, y n+2 =y n +2 x n+1. (18) This is an exact algorith which can be derived as follows (for another approach, see [10]). We consider the rotation fro the point x n, y n to x n+2, y n+2 ;seefigure 2.Hence,weobtain the following equations: x n+1 =x n cos ε y n, x n+2 =x n+1 cos ε y n+1, y n+2 =y n+1 cos ε+x n+1. Fro (20)weobtainvaluesx n+1 and y n+1 : x n+1 = y n+2 y n+1 cos ε, y n+1 = x n+2 x n+1 cos ε. y n+1 =y n cos ε+x n, (19) Substituting values given by (21)into(20)weget (20) (21) x n+2 2x n+1 cos ε+x n =0, (22) y n+2 2y n+1 cos ε+y n =0. Here we recognize the exact discretization of the classical haronic oscillator equation [16, 17]. Fro (22) itfollowsthat x n+1 = x n+2 +x n 2 cos ε, y n+1 = y n+2 +y n 2 cos ε. (23)

4 4 Modelling and Siulation in Engineering Table 2: Maxiu values of the absolute radial error (R =2 2+3 ). Nuber of algoriths =3 =5 =6 =8 =10 =12 (5) e e e e 2 (6) e e e e e 3 (7) e e e e 2 (11) 0.78e e e e e e 11 Substituting obtained values in (21), we get y x n+1 = y n+2 ((y n+2 +y n )/2cos ε) cos ε, y n+1 = x n+2 ((x n+2 +x n )/2cos ε) cos ε (24) y n+2 x n+2, y n+2 which can easily be reduced to y n+1 x n+1, y n+1 x n+1 = y n+2 y n 2, y n+1 = x n+2 x n 2, (25) that is, to (18). The algorith works with initial conditions: x 0 =R, y 0 =0, x 1 =Rcos ε,andy 1 =R,whereR=2 ( is an integer). Approxiating εone can transfor (18) into another two-step algorith: x n+2 =x n 2 ε y n+1, y n+2 =y n +2 ε x n+1, (26) y n ε ε x n, y n with x 0 =R, y 0 =0, x 1 =R 1 ε 2, y 1 =R.Thisalgorith was first proposed in [9]; see also [10]. Functional diagra of interpolator which reproduces (26) isshowninfigure 3. Interpolator contains 6 registers (Register 1, Register 2, Register 3, Register 4, Output x n+2,, and Output y n+2 ), 2 ultipliers (Right Shift Register 1 Right Shift Register 2), subtracter and adder. In this schee, at the beginning (before the interpolator starts) values of x 0 and y 0 arerecordedinregister2and Register 4, respectively, while values of x 1 and y 1 are recorded in Register 1 and Register 3. Then, values of x n+1 and x n are stored in Register 1 and Register 2, and values of y n+1 and y n. are stored in Register 3 and Register 4. Right Shift Register 1 and Right Shift Register 2 perfor ultiplication of values y n+1 and x n+1 by 2ε. Thus, at outputs of subtractor and adder parallel binary codes are fored in accordance with (26). Rounded values are recorded in Output x n+2 and Output y n+2. Table 3 contains axiu values of absolute radial systeatic error of proposed algorith, given by (26), after full circle siulation in the case d = 24, d = 32, ε=2, R = 2, =3,...,10, n=2π 1 ε, (27) where d is structure eleents datapath (in bits) and n is nuber of iterations. Table 4 contains the sae data for case R=2 2+3.NotethatresultsgiveninTables1 and 2 do not include round-off errors; hence they do not depend on d. x n+2 x n+1 Figure 2: Generation of 3 subsequent points along the circle. + Register 1 Register 2 Round Right Shift Register 1 x n Right Shift Register 2 x n+2 y n+2 Output Output Round x n+1 y n+1 x n Register 3 Register 4 Figure 3: Functional diagra for the circular interpolator (26). Analogical data for algorith given by (11) aregivenin Table 5 (we consider, as before, case (27)). Table 6 contains thesaedataforr= Coparing errors in Tables 3 and 5, andintables4 and 6, we see that the absolute radial errors of our algorith (26) (Tables 3 and 4) are saller than errors of well-known algorith (11) (see Tables 5 and 6). For exaple, we can copare values of errors at R=2 ( = 10) and d=24given in Tables 3 and 5. Consequently,wewillgetthatanabsolute + y n + + x

5 Modelling and Siulation in Engineering Figure 4: Nuber of correct bits of result as a function of at R= 2 and d=24. Table 4: Maxiu values of the absolute radial error (R =2 2+3 ) for proposed Algorith (26). d = 24 d = e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Figure 5: Nuber of correct bits of result as a function of at R= and d=24. Table 3: Maxiu values of the absolute radial error (R =2 )for proposed Algorith (26). d = 24 d = e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 7 radial error is saller by (ax(table 5))/(ax(Table 3)) = (1.267e 5)/(6.792e 6) = ties. We also see that the saller is, the greater is the difference between the corresponding errors of both algoriths. For exaple, for ( = 3) and d = 32 we obtain that an absolute radial error is saller by (ax(table 6))/ (ax(table 4)) = (7.781e 4)/(4.533e 10) ties. Finally, Figures 4, 5, 6, and7 present nubers of correct bits as a function of, for R=2 and R=2 2+3 and for d=24and d=32,respectively Figure 6: Nuber of correct bits of result as a function of at R= 2 and d=32. Table 5: Maxiu values of the absolute radial error (R =2 )for Algorith (11). d=24 d= e e e e e e e e e e e e e e e e e e e e e e e e e e e e 7 Presented figures clearly show that the proposed algorith (26) yields greater nuber of correct bits in coparison to algorith (11). 4. Conclusion A new circle generation DDA algorith has been proposed. In ters of hardware ipleentation and precision of circle generation the proposed DDA algorith for circle interpolator is ore efficient as copared with existing algoriths. We hopethattheproposedalgorithwillfindsoeapplications

6 6 Modelling and Siulation in Engineering Figure 7: Nuber of correct bits of result as a function of at R= and d=32. Table 6: Maxiu values of the absolute radial error (R =2 2+3 ) for Algorith (11). d = 24 d = e e e e e e e e e e e e e e e e e e e e e e e e 7 in coputer graphics (display screens), coputer-controlled printing, autoated control, and so forth. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. [5] X.Y.Fan,Y.H.Guo,andS.C.Li, Newdigitaldifferentialanalyzer interpolation algorith, Advanced Science Letters,vol.6, no. 1, pp , [6] X. L. Yan, W. L. Wang, and N. Xue, Research and siulation of an efficient circular interpolation algorith, Advanced Materials Research,vol ,pp ,2012. [7] O. Masory and Y. Koren, Reference-word circular interpolators for CNC systes, Engineering for Industry, vol.104, no. 4, pp , [8] Y.Katayaa,H.Harasaki,andM.Yano, Parallelalgorithof three diensional digital differential analyzer, in Proceedings of the Annual Convention ITE Japan, pp , [9] L. V. Moroz, A ethod for studying characteristics of digital sine-cosine generators, Inforation Extraction and Processing, vol.36,no.112,pp.84 90,2012(Ukrainian). [10] J. L. Cieśliński and L. V. Moroz, Fast exact digital differential analyzer for circle generation, [11] W. M. Newan and R. F. Sproull, Principles of Interactive Coputer Graphics, McGraw-Hill, [12] F. S. Li, Y. S. Wong, and M. Rahan, Circular interpolators for nuerical control: a coparison of the odified DDA techniques and an LSI interpolator, Coputers in Industry,vol. 18,no.1,pp.41 52,1992. [13] R. F. Eschenbach and B. M. Oliver, An efficient coordinate rotation algorith, IEEE Transactions on Coputers,vol.27,no.12, pp , [14] H. Hasegawa and Y. Okada, Apparatus and ethod for controlling a oving vehicle utilizing a digital differential analysis circuit, US Patent , [15] C. Pokorny Coputer Graphics, An Object Oriented Approach to the Art and Science, Franklin, Beedle & Associates Publishing, [16] J. L. Cieśliński and B. Ratkiewicz, On siulations of the classical haronic oscillator equation by difference equations, Advances in Difference Equations, vol.2006,articleid40171, [17] J. L. Cieśliński, On the exact discretization of the classical haronic oscillator equation, Difference Equations and Applications, vol. 17, no. 11, pp , Acknowledgent The second author (Jan L. Cieśliński) is partly supported by the National Science Centre (NCN) Grant no. 2011/01/ B/ST1/ References [1] W. Wang and C. Wang, Difference ethod for generation of circular arcs and ellipses, Coputer-Aided Design, vol. 21, no. 1, pp , [2] N. Matsushiro, A new digital differential analyzer for circl generation, IEICE Transactions on Inforation and Systes, vol. E81-D, no. 2, pp , [3] I. Chai, A high precision digital differential analyzer for circular interpolation, Tishreen University Journal, vol.29,no. 1, [4] H. Haa and T. Okuoto, Fast transfor into polar coordinate by generalized DDA, JournaloftheInstituteofTelevision Engineers of Japan,vol.47,no.1,pp ,1993.

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