In this article, the best uniform approximation for the hyperbola of degree 6 that has approximation order 12 is found. The associated error function vanishes 12 times and equioscillates 13 times. For an arc of the hyperbola, the error is bounded by 2:4 x 10-4. We explain the details of the derivation and show how to apply the method. The method is simple and this encourages and motivates people working in CG and CAD to apply it in their works

Rababah, Abedallah

Rababah, Esra’a

English

Institute of Advanced Engineering and Science

International Journal of Electrical and Computer Engineering (IJECE)Vol. 10, No. 2, April 2020, pp. 21922199ISSN: 2088-8708, DOI: 10.11591/ijece.v10i2.pp2192-2199 r 2192The best sextic approximation of hyperbolawith order twelveAbedallah Rababah1, Esra’a Rababah21Department of Mathematical Sciences, United Arab Emirates University, Al Ain, UAE1,2Department of Mathematics, Jordan University of Science and Technology, JordanArticle InfoArticle history:Received Apr 8, 2019Revised Oct 20, 2019Accepted Nov 2, 2019Keywords:approximation orderequioscillationhigh accuracyhyperbolasextic parametricapproximationABSTRACTIn this article, the best uniform approximation for the hyperbola of degree 6 thathas approximation order 12 is found. The associated error function vanishes 12times and equioscillates 13 times. For an arc of the hyperbola, the error is boundedby 2:4  10 4. We explain the details of the derivation and show how to applythe method. The method is simple and this encourages and motivates people work-ing in CG and CAD to apply it in their works.Copyright c 2020 Insitute of Advanced Engineeering and Science.All rights reserved.Corresponding Author:Abedallah Rababah,Department of Mathematical Sciences,United Arab Emirates University,Al Ain, UAE.Department of Mathematics,22110 Irbid, Jordan.Email: rababah@uaeu.ac.ae and esrababah15@sci.just.edu.jo1. INTRODUCTIONBe´zier curves were invented by Pierre Be´zier and Paul de Casteljau in the 1960’s. The primaryapplication was in the automobile industry, but today they are widely used in many computer applicationsand codes to sketch curves. Approximation using polynomials of low degree is favourable, and in manycases is a must issue. Many CAD systems are limited to using only parametric polynomial curves of lowdegree with low errors. The big error causes two major disadvantages: high accumulated error, and slow andcostly software. So, it is favourable to approximate using low degree polynomials. Parabolas are representedexactly using parametric curves of degree 2. In [1] methods of best uniform approximation of a circular arcof degree 6 with order 12 are accomplished. These approximations are optimal and have 13 equioscillations.In [2], a method of best uniform approximation of a hyperbola of degrees 2 with order 4 is accomplished.These results were motivated by the results of approximating curves with high accuracy [3-6].In this paper, we find approximation for the hyperbola of degree 6 and we get the order 12. Be´ziercurve techniques are used to represent the approximations of the hyperbola using parametric polynomials ofdegree 6 that have the least uniform error. The method is simple and practical. To achieve this approximation,proper arrangements and symmetries of the hyperbola are applied to determine the Be´zier points that definethe approximation of the hyperbola, and therefore, makes a CAD system more efficient and minimizes thecost. Using suitable translation and scaling, the hyperbola can be written in the basic forms: y2   x2 = 1 andx2   y2 = 1. Every form has two branches. So, there are four branches. Geometrically, all of these branchesJournal homepage: http://ijece.iaescore.com/index.php/IJECEInt J Elec & Comp Eng ISSN: 2088-8708 r 2193are identical. Therefore, it is sufficient to represent one branch and the other three branches can be representedusing rotation of this branch. So, we consider the upper branch of the hyperbola y2 x2 = 1, see Figure 1. It canbe written in parametric form c : t 7! (sinh(t); cosh(t)) ; t 2 <. We are interested in finding out the longestarc of the hyperbola that can be approximated and that the error function is the Chebyshev polynomial, see [2].It is impossible to exactly represent a hyperbola with a polynomial curve [7-11]. It can be represented exactlyusing rational Be´zier curves, a polynomial parametric form is preferred in many applications. The ability torepresent a primitive hyperbola is a must issue especially in computer graphics and data and image processing.Thus, there is a demand to find a parametrically defined polynomial curve p6 : t 7! (x6(t); y6(t)) ; 0  t  1,where x6(t); y6(t) are polynomials of degree 6. The p6 has to approximate c within tolerable error. In thispaper, degree 6 parametric curves are considered, and it is shown that the error is very small and the resultsare competitive.-20 -10 10 20-20-101020Figure 1. The Hyperbola y2   x2 = 1The Euclidean error function is used to measure the error between p6 and c. The error is defined by :E(t) :=qy26(t)  x26(t)  1: (1)E(t) is replaced by the following error function:e(t) := y26(t)  x26(t)  1: (2)The approximation problem is formulated as follows.The approximation problem is to find p6 : t 7! (x6(t); y6(t)) ; 0  t  1, where x6(t); y6(t) are polynomialsof degree 6, that approximates c by satisfying the following three conditions:(a) p6 minimizes maxt2[0;1] je(t)j,(b) p6 approximates c with order twelve [12],(c) e(t) equioscillates thirteen times over [0; 1].The solution to this problem is shown in section 3. This solution is presented in Figure 2and Figure 3; the corresponding error is shown in Figure 4. More related results can be found in [13-15]and the references therein.The best sextic approximation of... (Abedallah Rababah)2194 r ISSN: 2088-870821-21 21Figure 2. The Hyperbola and the sexticapproximating Be´zier curveFigure 3. Zoom in the busy part of Figure 20.2 0.4 0.6 0.8 1.0t-0.0004-0.00020.00020.0004eHtLFigure 4. The error of the sextic approximating Be´zier curve2. RESEARCHMETHODLet p6(t) = (x6(t); y6(t)) be a sextic polynomial parametric representation of the curve c. In CAGD,curves are presented using the Be´zier form, see Figure 5 for possible choice of the Be´zier points.P0P1P2P3P4P5P621-21 21Figure 5. Possible Be´zier points of the hyperbolaThe Be´zier curve p6(t) has the following form:p6(t) =6Xi=0piB6i (t) =:x6(t)y6(t); 0  t  1; (3)where p0; p1; p2; p3; p4; p5; p6 are the Be´zier points, and the Bernstein polynomial basis of degree 6 is definedby:B6i (t) =6i(1  t)6 iti; i = 0; : : : ; 6; t 2 [0; 1]:Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 2192 – 2199Int J Elec & Comp Eng ISSN: 2088-8708 r 2195The simplicity of this method should encourage people working in the fields of Computer Graphics,Image Processing, CAD, and Data Processing to adopt it in their design and applications. The symmetry inthe hyperbola is used to better locate the Be´zier points. We begin by letting p0 = ( ; ). The hyperbolais symmetric around the y-axis, so, to obey this symmetry, the point p6 should have the form p6 = (; ).The point p1 = ( ; ), so, to obey the symmetry, the point p5 should have the form p5 = (; ). The pointp2 = ( ;  ), so, to obey the symmetry, the point p4 should have the form p4 = (;  ). There is one remainingpoint; if this point lies in either halves of the plane around the y-axis, then the symmetry of the hyperbola iskicked. Thus, the point p3 must lie on the y-axis and has the form p3 = (0; !). Therefore, the proper choicefor the Be´zier points isp0 =  ; p1 =  ; p2 =   ; p3 =0!; p4 = ; p5 =; p6 =: (4)The Be´zier polynomial curve p6(t) in (3) is given in the form0@ x6(t)y6(t)1A =0@  B60(t)  B61(t)  B62(t) + B64(t) + B65(t) + B66(t)B60(t) + B61(t) +  B62(t) + !B63(t) +  B64(t) + B65(t) + B66(t)1A ; (5)where 0  t  1. The seven parameters ; ; ; ; ;  ; ! are used to have the polynomial approximationp6 comply with the conditions of the approximation problem; this is done in the following section.3. RESULTS AND ANALYSISThe values of ; ; ; ; ;  ; ! that minimize the uniform error and satisfy the conditions of theapproximation problem are given in the following theorem. A symbolic programming language is used to getthe values of the parameters in (4) and are rounded in decimal form.Theorem 1: The Be´zier curve in (5) with the Be´zier points in (4), where = 21:396696163346007;  = 21:420062908119476;  =  0:34812943434024657; =  0:3015887594987887;  = 0:5937616113806532;  = 0:705589524343983;! = 0:1813438330271954 (6)satisfies the three conditions of the Approximation Problem. More precisely, the error functions satisfy:  1211 e(t)  1211;   1211(2  )  E(t) 1211(2 + ); where  = max0t1jE(t)j: (7)Proof: From equation (5), we getx6(t) =  B66(t) B60(t)+  B65(t) B61(t)+  B64(t) B62(t);y6(t) =  B66(t) +B60(t)+  B65(t) +B61(t)+   B64(t) +B62(t)+ !B63(t):Substituting x6(t) and y6(t) into the error function e(t) in (2) and rewriting the result in termsof powers of t we get the following equality:e(t) = t12 42   48 + 1442 + 120   720 + 9002   80! + 480!   1200! + 400!2+t11  242 + 288   8642   720 + 4320   54002 + 480!   2880! + 7200!   2400!2+t10  362 + 288   5762   360 + 1440   9002 + 962   1032 + 27362+2400   12600 + 14400 2   1560! + 8160!   18600 ! + 6000!2+t9 1802   1440 + 28802 + 1800   7200 + 45002   2602 + 2520   57602The best sextic approximation of... (Abedallah Rababah)2196 r ISSN: 2088-8708 5400 + 23400   22500 2 + 3400!   14400! + 27000 !   8000!2+t8  4652 + 3480   64802   4170 + 15480   92252 + 5252   4440+86402 + 8430   29880 + 22725 2   5040! + 17040!   24000 ! + 6000!2+t7 7802   5280 + 86402 + 5880   18720 + 99002   8042 + 5808   95042 9240 + 26640   15300 2 + 5040!   13440! + 13200 !   2400!2+t6  9222 + 5460   74882   5460 + 13860   58502 + 9262   5628+76322 + 7140   16380 + 6750 2   3360! + 6720!   4200 ! + 400!2+t5 7922   3948 + 43202 + 3360   6300 + 18002   7922 + 3972   43202 3840 + 6660   1800 2 + 1440!   1920! + 600 !+t4  4952 + 1980   16202   1320 + 1620   2252 + 4952   1980 + 16202+1380   1620 + 225 2   360! + 240!+t3 2202   660 + 3602 + 300   180   2202 + 660   3602   300 +180 + 40!)+t2  662 + 132   362   30 + 662   132 + 362 + 30 +t 122   12   122 + 12+ ( 1  2 + 2):We substitute the values of ; ; ; ; ;  and ! from (6) into the last equation to gete(t) = 4096 t12   24576 t11 + 64512 t10   97280 t9 + 93024 t8   58752 t7 + 24752 t6 6864 t5 + 1930516t4   10018t3 +42964t2   964t+12048; t 2 [0; 1]:Making the substitution t = u+12 reduces the error function to the following forme(u) =12048  9256u2 +105256u4   74u6 +278u8 + 3 t10 + t12; u 2 [ 1; 1]:The last form of e(u) coincides with the monic Chebyshev polynomial ~T12(u); u 2 [ 1; 1], which is theunique polynomial of degree 12 that equioscillates 13 times between  1211 for all u 2 [ 1; 1] and has the leastdeviation [12]. This shows that p6 satisfies the conditions of the approximation problem. The error formula forE(t) can be proved using its relation to the error function e(t). This proves Theorem 1.It is clear that the approximation is the best uniform approximation from Figure 4. Figure 2 and Figure 3 showthe hyperbola and the approximating Be´zier curve, Figure 4 shows the corresponding error, and Figure 6 showsthe Euclidean error.4. CONCLUSIONIn this paper, the best uniform approximation of the hyperbola with parametrically defined polyno-mial curves of degree 6 are explicitly given. The error function equioscillates 13 times; the approximationorder is 12. The method of construction demonstrates the efficiency and simplicity of the approximationmethod. The approximation intersects the hyperbola 12 times with maximum error 2:4 10 4. Reflecting theupper branch of the hyperbola around the x-axis gives the lower branch. The hyperbola is shown in Figure 7.Note that the points p1 (p5) and p2 (p4) in both branches are very close to each other and can not be distin-guished from each other. The results in this paper can be used to improve the results obtained in [16-23] seealso the results in [24, 25].Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 2192 – 2199Int J Elec & Comp Eng ISSN: 2088-8708 r 21970.2 0.4 0.6 0.8 1.0t-0.0002-0.00010.00010.0002EHtLFigure 6. The Euclidean error of the sextic Be´zier curve21-21 21-21Figure 7. Both branches of the Hyperbola using two Be´zier curvesACKNOWLEDGMENTSThe authors owe thanks for the reviewers for invaluable suggestions to improve an earlier version ofthis paper. This research was funded by Jordan University of Science and Technology.REFERENCES[1] A. Rababah, ”Best sextic approximation of circular arcs with thirteen equioscillations,” Pro-ceedings of the Jangjeon Mathematical Society, vol. 21, Issue 1, pp. 111–123, 2018. DOI:http://doi.org/10.17777/pjms2018.21.1.111[2] A. Rababah, ”The Best Quadratic Approximation of Hyperbola with Order Four,” Journal of Inequalitiesand Special Functions, vol. 7, Issue 3, pp. 1–11, 2016.[3] M. Floater, ” High order approximation of conic sections by quadratic splines,” Computer Aided Geomet-ric Design, vol. 12, pp. 617–637, 1995.[4] C. de Boor., et al., ”High accuracy geometric Hermite interpolation,” Computer Aided Geometric Design,vol. 4, pp. 269–278, 1988.[5] A. Rababah, ”Taylor theorem for planar curves,” Proceedings of the American Mathematical Society,vol. 119, Issue 3, pp. 803–810, 1993.[6] A. Rababah, ”High accuracy Hermite approximation for space curves in <d,” Journal of MathematicalAnalysis and Applications, vol. 325, Issue 2, pp. 920–931, 2007.[7] K. Ho¨llig and J. Ho¨rner, ”Approximation and Modeling with B-Splines,” SIAM, Titles in Applied Math-ematics, vol. 132, 2013.[8] H. Prautzsch., et al., ”Be´zier and B-Spline Techniques,” Springer 2002.The best sextic approximation of... (Abedallah Rababah)2198 r ISSN: 2088-8708[9] P. Be´zier, ”The mathematical basis of the UNISURF CAD system,” Butterworth-Heinemann Newton,MA, USA, ISBN 0-408-22175-5, 1986.[10] Carl de Boor, A Practical Guide to Splines, Series Title: Applied Mathematical Sciences, Vol 27, Springer-Verlag New York, 1978.[11] Gerald Farin, Curves and surfaces for computer-aided geometric design: a practical guide. Elsevier, 2014.[12] J. Rice, ”The approximation of functions,” linear theory, Addison-Wesley, vol. 1, 1964.[13] Y. J. Ahn and C. Hoffmann, ”Circle approximation using LN Be´zier curves of even degree and its ap-plication,” Journal of Mathematical Analysis and Applications, vol. 410 Issue 1, pp. 257–266, February2014.[14] S. W. Kim and Y. J. Ahn, ”Circle approximation by quartic G2 spline using alternation of error function,”J. KSIAM, vol. 17, Issue 5, pp. 171–179, 2013.[15] A. Rababah., et al., ”Multiple Degree Reduction and Elevation of Be´zier Curves Using Jacobi-BernsteinBasis Transformations,” Numerical Functional Analysis and Optimization, vol. 28, Issue 9-10, pp. 1179–1196, 2007.[16] S. Alabed, ”Performance Analysis of Differential Beamforming in Decentralized Networks,” Interna-tional Journal of Electrical and Computer Engineering (IJECE), vol. 8 Issue 3, pp. 1692-1700, 2018.DOI: http://doi.org/10.11591/ijece.v8i3.pp1692-1700[17] C. Suwannapong and C. Khunboa, ”An Approximation Delay between Consecutive Requests for Con-gestion Control in Unicast CoAP-based Group Communication,” International Journal of Electrical andComputer Engineering (IJECE), vol. 9, Issue 3, June 2019. DOI: http://doi.org/10.11591/ijece.v9i3.[18] Bhargav Bhatkalkar, Abhishek Joshi, Srikanth Prabhu, Sulatha Bhandary, Automated fundus image qual-ity assessment and segmentation of optic disc using convolutional neural networks. International Journalof Electrical and Computer Engineering (IJECE) Vol.10, No.1, 2020, pp. 816 827, ISSN: 2088-8708,DOI: 10.11591/ijece.v10i1.pp816-827.[19] Rodiah, Sarifuddin Madenda, Diana Tri Susetianigtias, Dewi Agushinta Rahayu, EtySutanty, Optic Discand Macula Localization from Retinal Optical Coherence Tomography and Fundus Image, InternationalJournal of Electrical and Computer Engineering (IJECE) Vol.8, No.6, 2018, pp. 5050-5060, ISSN: 2088-8708, DOI: 10.11591/ijece.v8i6.pp5050-5060.[20] G. Daway, Hana H. kareem, Ahmed Rafid Hashim, Pupil Detection Based on Color Difference and Circu-lar Hough TransformHazim, International Journal of Electrical and Computer Engineering (IJECE) Vol.8,No.5, 2018, pp. 3278-3284, ISSN: 2088-8708, DOI: 10.11591/ijece.v8i5.pp3278-3284.[21] A. Rababah, L-2 degree reduction of triangular Be´zier surfaces with common tangent planes at ver-tices. International Journal of Computational Geometry & Applications, Vol. 15, No. 5 (2005) 477-490.http://www.worldscinet.com/ijcga/15/1505/S02181959051505.html[22] A. Rababah, Distances with rational triangular Be´zier surfaces. Applied Mathematics and Computation160, (2005) 379-386. http://www.elsevier.com/locate/amc[23] A. Rababah and Moath Jaradat, Approximating offset Curves using Be´zier curves with high accuracy,International Journal of Electrical and Computer Engineering (IJECE) Vol.10, No.2, 2020 (PART I).[24] A. Rababah, ”Approximation von Kurven mit Polynomen und Splines,” Ph. Dissertation, Stuttgart Uni-versita¨t, West Germany, 1992.[25] A. Rababah, The best quintic Chebyshev approximation of circular arcs of order ten, International Journalof Electrical and Computer Engineering (IJECE)Vol. 9, No. 5, October 2019, pp. 3779-3785, ISSN: 2088-8708, DOI: 10.11591/ijece.v9i5.pp3779-3785.Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 2192 – 2199Int J Elec & Comp Eng ISSN: 2088-8708 r 2199BIOGRAPHIES OF AUTHORSAbedallah Rababah is a professor of mathematics at United Arab Emirates University and is onleave from Jordan University of Science and Technology. He is working in the field of ComputerAided Geometric Design, abbreviated CAGD. In particular, his research is on degree raising andreduction of Be´zier curves and surfaces with geometric boundary conditions, Bernstein polynomials,and their duality. He is known for his research in describing approximation methods that significantlyimprove the standard rates obtained by classical (local Taylor, Hermite) methods. He proved the fol-lowing conjecture for a particular set of curves of nonzero measure: Conjecture: A smooth regularplanar curve can, in general, be approximated by a polynomial curve of degree n with order 2n.The method exploited the freedom in the choice of the parametrization and achieved the order 4n/3,rather than n + 1. Generalizations were also proved for space curves. Professor Rababah is also do-ing research in the fields of classical approximation theory, orthogonal polynomials, Jacobi-weightedorthogonal polynomials on triangular domains, and best uniform approximations. Since 1992,He has been teaching at German, Jordanian, American, Canadian, and Emirates’ universities.He is active in the editorial boards of many journals in mathematics and computer science.Further info can be found on his homepage at ResearchGate:https://www.researchgate.net/profile/Abedallah Rababahor at http://www.just.edu.jo/eportfolio/Pages/Default.aspx?email=rababahEsra’a Rababah graduated from Jordan University of Science and Technology. She obtained bothBachelor and Master Degrees in Mathematics with research interests in the field of numerical analy-sis, approximation theory, and Be´zier techniques. She worked as teacher assistant at Jordan Univer-sity of Science and Technology. Further info can be found on her homepage at ResearchGate:https://www.researchgate.net/profile/Esraa RababahThe best sextic approximation of... (Abedallah Rababah)

The best sextic approximation of hyperbola with order twelve

In this article, the best uniform approximation for the hyperbola of degree 6 that has approximation order 12 is found. The associated error function vanishes 12 times and equioscillates 13 times. For an arc of the hyperbola, the error is bounded by 2:4  10 4 . We explain the details of the derivation and show how to apply the method. The method is simple and this encourages and motivates people working in CG and CAD to apply it in their works

The best sextic approximation of hyperbola with order twelve

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