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World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015 Circular Approximation by Trigonometric Bézier Curves Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225 Abstract—We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques. Keywords—Control points, rational trigonometric Bézier curves, radius error, shape measure, weight functions. I. INTRODUCTION C IRCLES are around us everywhere and inspired the mankind even before the beginning of recorded history. Circles are the main source of many wonderful inventions, such as circular ripples, wheels, circular gears which run the machines in urban factories and make our life much easier. Circular arcs are basic tools in engineering design, web design and mobile design. They are also used as user interface tool and in many other projects due to their exceptional properties such as constant curvature, constant distance from a fixed point etc. The trigonometric polynomial curves were first introduced by Schoenberg [6]. The control point form of quadratic and cubic trigonometric polynomial curves were presented by Han in [3] and [4]. Wang, Chen and Zhou [7] defined trigonometric B-splines, known as algebraic-trigonometric Bsplines (NUAT). These contributions only narrowed down the properties of trigonometric polynomial curves and did not focus on its applications. The authors in [7] used trigonometric polynomial curves for the shape preservation of data. In this research paper, the circular approximation problem is considered. This problem is defined as: Given two end points and end tangents/unit tangents of a circular arc, construct a curve which interpolates the end points of circular arc and close to it between these endpoints. Integral Bézier curves were in account for the approximation of circular arc by a number of researchers. Lee , and [5], Goldapp [2] and Fang [1] presented the approximations of circular arc by quadratic, cubic and quintic Bézier curves respectively. The radius error between the circular arc and the approximating Bézier was minimized to obtain the best approximation. In this research paper, rational cubic trigonometric Bézier curves are developed for approximating circular arc, particularly when its 1. Two end points and end tangents vectors are given. 2. Two end points and end unit tangents vectors are given. In this section, rational quadratic and cubic trigonometric Bézier curves are introduced over the interval 0, . The parametric form of newly developed rational quadratic , 0, , is defined by the trigonometric Bézier curve relation given in (1): ∑ , ∑ (1) where and are the quadratic trigonometric basis are the functions and weight functions respectively. , , control points of rational quadratic trigonometric Bézier curve. The quadratic trigonometric basis functions of (1), are defined as 1 2 1 , . , It can be easily verified that ∑ 1 and 0 for 0,1,2, (2) where , ∑ and 0 , . (3) Thus from (2) and (3), the rational quadratic trigonometric Bézier curve (1), satisfies the convex-hull property and endpoints interpolation property. The parametric form of the developed rational cubic trigonometric Bézier curve is given by the following relation. Maria Hussain is an Assistant Professor in Department of Mathematics, Lahore College for Women University, Lahore, Pakistan (phone: +92-03314930071; e-mail: [email protected]). Malik Zawwar Hussain is a Professor in Department of Mathematics, University of the Punjab, Lahore, Pakistan (e-mail: [email protected]). Mubashrah Saddiqa is a Lecturer in Department of Mathematics, Lahore College for Women University, Lahore, Pakistan. International Scholarly and Scientific Research & Innovation 9(1) 2015 RATIONAL TRIGONOMETRIC BÉZIER CURVE II. ∑ ∑ 30 ,0 scholar.waset.org/1999.7/10000225 , (4) World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015 and are the control points, weight functions where , and basis functions respectively. These cubic trigonometric , are defined as basis functions 3 1 , 0,1,2,3. International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225 1, 0 for 0,1,2,3,0 , 1 , . 0,1,2, and , 2 , , cos . , where (8) Thus the rational cubic trigonometric Bézier curve is rewritten as . , , , The weight functions positive real numbers. , (9) where control points , 0,1,2,3 and weight functions , are defined in (8). With reference to Fig. 1, the following notations are introduced where / , (5) The cubic trigonometric Bézier curve (4) satisfies the following properties: and , (i) End point interpolation property: 0 (ii) Convex-hull property: ∑ rational quadratic trigonometric Bézier curve (7) and rational cubic trigonometric Bézier curve (4) is given as follows . (10) 0,1,2,3, are It can be seen from Fig. 1 that cos , or . Also, putting the values of and from (7) in (10) to get . Since cos , so can be written as a function of as . (11) By vector algebra, from Fig. 1, we have Fig. 1 The control polygon of rational quadratic and rational cubic Bézier curve . (12) From (10) and (11), we get III. REPRESENTATION OF A CIRCLE BY TRIGONOMETRIC BÉZIER CURVES cos In this section, the rational quadratic and rational cubic representations of a circular arc are established. If the control form an isosceles triangle with base and points , , 0,1,2 taken as the weight functions , 1, cos , (7) The control point is the intersection of the lines given by the end points , along with unit tangents , at and respectively as shown in Fig. 1. Now by degree elevation of rational quadratic trigonometric Bézier curve (7), we get a rational cubic trigonometric representation of the same and weights circular arc with control points , , and 1, , , 1. The relation between control points and weights of International Scholarly and Scientific Research & Innovation 9(1) 2015 (13) . IV. RATIONAL CUBIC TRIGONOMETRIC APPROXIMATION OF CIRCLE WHEN TWO END POINTS AND END TANGENTS ARE GIVEN (6) then the rational quadratic trigonometric Bézier curve (1) represents circle. Here is the base angle of the triangle (See Fig. 1). Thus the rational quadratic trigonometric Bézier curve (1) is rewritten as . Similarly, . In this section, the problem of approximation of a circle by rational cubic trigonometric Bézier curve (9) is discussed. If two end points and end tangents of a circular arc are given, then the end control points and of the rational cubic trigonometric Bézier curve (9) are equal to the end points of the given circular arc. For solving this problem, we calculate , and the corresponding weights the control points , . From Fig. 1, we have , where and are the end tangents, given as . 31 , scholar.waset.org/1999.7/10000225 (14) and World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015 0 The derivatiive vectors 0 and 3 and 3 thhese derivativee vectors are rewritten r as 0 3 . Usingg (14), 3 and . This gives and , (15) w where International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225 1 2 cos 1 annd 2 cos . (16) is the anggle between 0 and and is thhe angle beetween and . Now, when n the input data is cooming from a circle, then the rational cubic trigonoometric Bézier curve, with w control points p and weeights definedd in (8), w reproduce the will t circle. V. RATIONALL TRIGONOMETTRIC CUBIC APPROXIMATIO V ON OF CIRCLE WHEN N TWO END POINTS O AND END N UNIT TANG GENTS AR RE GIVEN In this sectioon, the probleem of approxiimation of a circle c is diiscussed whenn its two endd points and end e unit tangeents, annd are giveen. Again thee end points of o the rationaal cubic triigonometric Bézier B curve are identical to the end pooints of thhe given circuular arc. Thus we have to fiind the Bézierr points and and the t weights and . Thee unit tangentss and make the angles and with the base respeectively. U Using (13), we can find the B Bézier points and i.e. , . (18) . | In this sectioon, a circular arc is estimaated by the scchemes developed in Seections IV andd V. (a)). Minimizatiion of : In this case the t values of and are obtaained by minim mizing the inntegral given in i (18) for rational cubic trigonoometric Bézier curve (9). The T end 1,0 1 and 1,0 0 . The points of thhe circle are taaken as optimized values of 1.6435an nd =1.47322, and correspondiing to these values we have h 1,1.6896 , 1,2.0350 0 , 0.3946 and 0..3276. The reesulting graph is cloose to circle with radius error e betweenn 0 and 0.0836. Thee approximateed circle is shoown in Fig. 2. (b)). Minimizatiion of : In this case the values of and are obtaained by minim mizing the inntegral given in i (19) 1,0 for (9). Thee end points oof the circle arre taken as and 1,0 . The optim mized values of 1.57008 and =1.5708, and correspoonding to theese values wee have 1,2 , 1,2 , 0.333 and 0.3333 3. The r error beetween resulting grraph is close too circle with radius 0 and 9.98880 10 . The T approximaated circle is shown in Fig. 3. (c)). Geometric Approach: When we used u the geoometric t circle is rreproduced wiith and d = approach, the (See Fig. 4). 4 Corresponnding to thesse values, wee have and . In this casse, the 1,2 , 1,2 , rational triggonometric cuurve (9) will reproduce r thee circle (See. Fig. 5) which iss the same as a obtained by b the minimizatioon of . In Figs. 6 and a 7 the currvature derivative plots p of (9) arre shown for the t above menntioned cases. (17) The correspponding weigght functionns and are caalculated by (16). The otherr weights are unity. The anngles annd are approximated byy minimizing the shape measure m quuantities and defined in (18) andd (19) reespectively. ′ VI. NUMER RICAL EXAMPL LES are compuuted as | VII. CONCLUSION In this research paper, we have discusseed the rational cubic triggonometric teechniques to approximate the t circle. We W have useed approachees, the variatioonal and the geometric onne. The quuality of thee rational cuubic trigonom metric probllem is callculated usingg (20), which is found to be b 17.6 6099. It is 375.3 for pollynomial Herm mite interpolaant it is 375.33. This r cubic trigonometricc schemes didd better shoows that our rational thaan integral and a rational polynomial p Hermite H interrpolant. Raadius error is found f less thann many existing schemes [11]. (19) Equation (188) shows thatt the change in i curvature is i more siggnificant than n its magnitudde. The disconntinuous curvature is accceptable as loong as the sloppe ′ is conntinuous. The quality off the develo oped rationaal cubic trig gonometric circular c appproximation scheme is measured by , defined byy , . F 2 A rationall cubic trigonom Fig. metric approxim mation of circlee using (20) mitations of addmissible values for Here, , , , are the lim and . Heree 90 , 90 and 90 , 270 i.e. alll acceptable have a positive -ccomponent, and a all addmissible haave a negativee -componennt. International Scholarly and Scientific Research & Innovation 9(1) 2015 32 scholar.waset.org/1999.7/10000225 World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015 [7] G. Wang, Q. Chenand M. Zhhou, “NUAT B-spline curves,”C Computer 2 2004, pp. 193--205. Aided Geomettric Design,vol. 21, International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225 F 3 A rationaal cubic trigonoometric approxim Fig. mation of circlee using Fig. 4 Tan ngents direction n Fig. 5 The graphical g represeentation of circcle using geomeetric ap pproach F 6 Curvaturee derivative plo Fig. ot of cubic trigonometric Bézieer using geomettric approach Fig. 7 Cu urvature derivative plot of cubiic trigonometricc v apprroach Bézier using variational REFFERENCES [1]] [2]] [3]] [4]] [5]] [6]] L. 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