On Parametric Polynomial Circle Approximation

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1 Numerical Algorithms manuscript No. will be inserted by the editor On Parametric Polynomial Circle Approximation Gašper Jaklič Jernej Kozak Received: date / Accepted: date Abstract In the paper, the uniform approximation of a circle arc or a whole circle by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too. Keywords circle arc parametric polynomial curve uniform approximation Mathematics Subject Classification A10 65D10 65D17 1 Introduction A circle arc is a basic object in CAGD. Unfortunately, it does not have a parametric polynomial representation. Since it can be represented by a quadratic rational Bézier curve, this was one of the main reasons that NURBS are nowadays standard objects in CAGD. Recently, in [19] the authors pointed out some of the main problems of using rational objects and argued, that by using fast iterative polynomial approximants, good results can be obtained and we could drop R from NURBS. This was demonstrated on an approximation of a circle arc. G. Jaklič FGG and IMFM, University of Ljubljana, Jamova, 1000 Ljubljana, Slovenia, and IAM, University of Primorska, Muzejski trg, 6000 Koper, Slovenia Tel.: Fax: gasper.jaklic@fgg.uni-lj.si J. Kozak FMF and IMFM, University of Ljubljana, Jadranska 1, 1000 Ljubljana, Slovenia

2 Gašper Jaklič, Jernej Kozak A natural question is, how good of a parametric polynomial approximation of a circle arc can one obtain? Quite a few papers study good approximation of circular segments with the radial error as the parametric distance. Quadratic Bézier approximants are considered in [18], and their generalizations to the cubic case can be found in [4] and [9]. The quartic case is systematically studied in [1], [15] and [10], and quintic Bézier approximants are derived in [5] and [6]. Recently, quartic G 1 approximants were analysed in [16]. General results on Hermite type approximation of conic sections by parametric polynomial curves of odd degree are given in [7] and [8]. The results hold true only asymptotically, i.e., for small segments of a particular conic section. Hermite approximation of ellipse segments by cubic parametric Bézier curves is studied in [3] and also in []. In recent years some quite surprisingly good approximations of the whole circle were obtained. An approach based on Taylor approximation was improved by the idea of geometric interpolation and a construction of polynomial approximants for all odd degrees was obtained in [17]. The construction that covered also even degrees was presented in [1]. By looking at the problem from a different perspective, it turned out that the obtained construction was just one of several solutions of a nonlinear problem, and that there exist better solutions. In [14], the best such solution was presented, which gives a good approximation of a conic section. It has many nice properties, it is symmetric, shape preserving, it gives a high order approximation of the whole circle, and for higher degrees it circles the circle several times before it deviates from it. Furthermore, it is given in a closed form. In [11], the result was improved, but the solutions were not given exactly. Our goal is to construct parametric polynomials x n and y n of degree n, which yield a good approximation of a circle arc that can be the whole unit circle Let us consider the expression cos θ + sin θ = 1. x nt + y nt = 1 + εt. 1 Here, ε is a polynomial of degree n. Since the circle does not have a parametric polynomial parameterization, every polynomial approximation x n, y n yields a deviation ε from the unit circle with the implicit equation x + y = 1. So, in some sense, ε should be small to obtain a good approximation. There are several issues to be answered: the choice of the error function radial error, parametric distance, Hausdorff distance,..., the choice of ε, how to obtain x n and y n from the equation 1.

3 On Parametric Polynomial Circle Approximation 3 Furthermore, the solution should resemble the circle arc and the error should be as small as possible. In [14], the choice εt = t n was analyzed and a nice approximation in a closed form was obtained, which enables precise error analysis and applications. Its disadvantage is, that it is a one-sided approximation of the circle, and has only one multiple contact with the circle at t = 0, and is thus Taylor-like in its nature. An idea right at hand would be to take x n and y n as Taylor expansion of cos and sin up to the degree n. This yields a good local approximation near 0, but the approximation quickly deviates from the circle [13]. In this paper, we analyse the case 1 + εt = 1 r 1 + at nt, r := ra := 1 1 a a, where 0 < a < 1 is an unknown parameter and T n denotes the Chebyshev polynomial of the first kind. Numerically, up to n = 9, the solutions to this problem were given in [11], but significant numerical instability was reported. Here, we overcome the shortcomings of that approach. A closed form of the solution derived in this paper helps us to provide a numerically stable construction as well as outline a detailed asymptotic analysis of its properties. The best solution as a function of a exists and it is unique. Fig. 1 Parametric polynomial approximation of degree 4 blue of a circle red. As a motivation, see Fig. 1, where such an approximant of degree 4 is shown. The radial error is , thus the curves are almost indistinguishable. The method can be extended to circle arcs. The radial error for a quartic approximation of a quarter of a circle is shown in Fig.. The approximant has very good approximation properties: n contacts, asymptotic approximation order decreases faster than exponentially, etc. The computation

4 4 Gašper Jaklič, Jernej Kozak Fig. Error at a quartic approximation of a quarter of a circle. of the optimal parameter a requires just a few steps of the tangent method. For n = 3, 4, the exact parameter is computed as a solution of a cubic and quadratic equation, respectively. The best parametric polynomial approximant is applied for a fast and stable evaluation of trigonometric functions. Best uniform circle approximation Let c τ : [ π τ, π τ ] R, θ c τ θ := cos θ, τ R, τ 1, sin θ denote a circle arc of the unit disk parameterized by the angle θ, and let c := c 1 denote the whole circle. We are looking for a parametric polynomial curve that in some sense well approximates c τ. In order to define the meaning of the word well, one has to choose a measure of distance between two curves. Let. denote the Euclidean norm in R, and let, more generally, q := q1 q : [α, β] R denote a continuous parametric curve. A popular choice of the distance measure between a circle arc and its approximation is the radial one, determined by the radial error function e, et := et; q := 1 q t, dist R c τ, q := max α t β et =: e. The radial distance between c τ and q is well defined if we restrict curves to admissible ones only. To be precise, a parametric curve q = q 1, q is an admissible approximation of a circle arc c τ if the argument map arg q 1 + i q : [α, β] [ π τ, π τ ]

5 On Parametric Polynomial Circle Approximation 5 is surjective. Let dist H c τ, q denote the Hausdorff distance between c τ and q, based upon the Euclidean norm. Quite clearly, dist H c τ, q dist R c τ, q. But in our particular case, we have the following observation. Lemma 1 Let q be an admissible approximation of a circle arc c τ. Then Proof dist H c τ, q = { = max max π/τ θ π/τ dist c τ, q := dist H c τ, q = dist R c τ, q. min c τ θ qt, max α t β α t β max min c τ θ qt α t β π/τ θ π/τ } min c τ θ qt π/τ θ π/τ = max α t β c τ arg q 1 t + i q t qt = e = dist R c τ, q, since the argument map gives the closest circle arc point to any qt. Let now q = p n, q 1 = x n, q = y n be a parametric polynomial curve of degree n, with x n, y n R[t] being polynomials of degree n. Without losing generality, we assume also [α, β] = [ 1, 1]. The radial error function could be written in a form et = 1 p n t = 1 x nt + y nt = a εt, 3 where a > 0 is a constant, and ε R[t] is a polynomial of degree n that satisfies ε 1 a on [ 1, 1]. In [11], the choice ε = T n as a tool to the whole circle approximation was studied thoroughly. It was also observed that such a polynomial does not produce an equioscillating error function. So a numerical procedure was proposed to improve the circle approximant. In particular, the whole circle approximation x n, y n was obtained from where ξ k := cos n 1 x nt + ynt = 1 + a T n t = a n 1 i n k=0 t ξ k, 4 πk n i 1 n ln a 1 1, k = 0, 1,..., n 1, 5 a with joining proper linear terms t ξ k together in order to obtain the factorisation x nt + y nt = x n t + i y n t x n t i y n t. 6 But a polynomial that leads to an equioscillating error function is not far from 4.

6 6 Gašper Jaklič, Jernej Kozak Lemma Let the polynomial ε be chosen as ε = 1 r ar + 1 r T n, r := ra := 1 1 a a. 7 The corresponding radial error function 3 on [ 1, 1] alternately reaches its extreme values ± e, e = max et = a 1 t 1, 8 1 a a at n + 1 extrema of T n. Proof Let ε be given by 7. Then et = atn t, e at t = nt r r 1 + at n t. 9 Since 1 T n 1 on [ 1, 1], 1 + a 1 a 1 + a 1 a et, t [ 1, 1], 1 + a + 1 a 1 + a + 1 a and the assertion follows. Observe from 8 that the radial error e = dist c τ, p n is an increasing function of a, d da e = 1 1 a + 1 a > 0. So the optimal circle arc parameterization will be obtained at the smallest value of a such that 1 + a ε is admissible. Following [11], we are looking for the conjugate factorisation 6 of the polynomial with 1 + aεt = 1 n 1 r 1 + at nt = γ i n γ := γn; a := 1 r Let us introduce a new constant k=0 t ξ k, 10 a n a n = a a 1 a z := zn; a := 1 + n > a Note that z = za is an increasing function of a, d zn; a zn; a = da na 1 a > 0.

7 On Parametric Polynomial Circle Approximation 7 So the smallest a requires the smallest z too. With a help of 1, the roots ξ k in 5 simplify to ξ k = z k+1π ei n 1 z e i k+1π n, k = 0, 1,..., n Observe that ξ k = ξ n 1 k. So if t ξ k is a linear divisor of the first factor of the right hand side of 6, t ξ n 1 k should divide the second one. Thus a candidate for the required split 6 of 10 is any polynomial of the form n 1 p n,σ t := x n,σ t+i y n,σ t := γ i t + i z σ ei k k+1π n + i 1 z e i σ k k+1π n, k=0 14 where x n,σ and y n,σ denote the real and the imaginary part of p n,σ, respectively, and σ := σ k n 1 k=0, σ k { 1, 1}. We observe that an admissible polynomial p n,σ should satisfy g σ z := arg x n,σ 1 + i y n,σ 1 = π τ. 15 Note that factors of p n,σ 1 in 14 could be rewritten as i + i z ei σ k k+1π n + i 1 1 z z e i σ k k+1π n = 1 + i z sin π k + 1σ k + n + z + i cos π k + 1σ k + n This gives the equation 15 in the form d dz arg n 1 k=0 But the derivatives arg 1 + i z sin π k + 1σ k + n + z + i cos π k + 1σ k + n = π τ. 1 + i z sin π k + 1σ k + n + z + i cos π k + 1σ k + n sin πk+1σ k n = z cos πk+1σ k n + z + 1, 0 k n 1, are positive if σ k = 1, and negative if σ k = 1. So the function g σ will have the steepest ascent if one chooses all σ k to be equal to 1, i.e., σ = 1. Let us consider 16 with such a choice. At z = 0, the sum of terms at k = l and k = n 1 l equals πl + n + 1 πl + n + 1 arg sin + i cos πl + n + 1 πl + n arg sin i cos = 0,. 16

8 8 Gašper Jaklič, Jernej Kozak and πk + n + 1 arg sin + i cos for n odd and k = n 1/, thus g 1 0 = 0. At z = 1, the terms in 16 simplify to πk + n + 1 arg 1 + i sin + cos πk + n + 1 = 0 πk + n + 1 = arg 1 + i = π 4. In order to observe this, note that for all k, 0 k n 1, we have sin πk + n cos πk + n + 1 = sin Thus g 1 1 = nπ. This proves the following theorem. πk + n + 1 > 0. Theorem 1 The smallest positive solution z, 0 < z 1, of the equation 15 is unique. It is determined by the choice σ k = 1, k = 0, 1,..., n 1. In order to make algorithms more efficient, we further simplify p n,1 and g 1. Let and s := 1 a = as = 1 z z 1 > 0, z = s s, g s := g 1 s s n s s + 1, 1 s s πk + 1 ν k := ν k,n := sin n With a switch from z to s we observe from 13 Thus, n > 0, k = 0, 1,..., 1. i t ξ k i t ξ n 1 k = 1 ν k,n + s t + i stν k,n. p n,1 t = x n,1 t + i y n,1 t = γqt n 1 k=0 where qt = s + i t if n is odd, and qt = 1, otherwise ν k,n + s t + i sν k,n t, Remark 1 From 18 we observe that x n,1 is an even and y n,1 an odd polynomial. This implies y n,1 0 = 0. Also, x n,1 0 > 0 if s > 0. 18

9 On Parametric Polynomial Circle Approximation 9 Factors in 18 at t = 1 simplify to 1 ν k,n + s t + i sν k,n t t=1 = s + i ν k,n. This allows one to write the equation 16 at σ = 1 as g s = n 1 k=0 µ k arctan νk,n s = π {, 0 k < n τ, µ k :=, 1, k = n 1 = n. 19 Theorem 1 implies that the equation 19 has a unique positive solution s = s. Thus 17 determines a := as, and γ := γn; a by 11. Let us denote the optimal solution as p nt := x nt, y nt := x n,1 t, y n,1 t s=s, γ=γ. 0 Theorem Let s > 0 be the solution of 19, and let p n be the corresponding parametric polynomial circle arc approximation defined in 0. The argument map, introduced in, arg x n+ i yn : [ 1, 1] [ π τ, π ], 1 τ is a diffeomorphism. Proof Recall 18. Note that d dt arctan sν k,n t 1 νk,n + s t = sν k,n 1 ν k,n + s + t 4s t ν k,n + 1 ν k,n + s t > 0 and d dt arctan t s = s s + t > 0. So the continuous function arg x n+ i yn is differentiable and monotonously increasing. By the choice of s it satisfies arg x n1 + i yn1 = g s = π τ. By Remark 1, one also has arg x n 1 + i y n 1 = arg x n1 i y n1 = g s = π τ, so the map 1 is bijective, which completes the proof. In [17], a parametric distance dist P.,.. between parametric curves was introduced as a distance measure that can usually be more easily estimated than the Hausdorff one. But Theorem proves that the map 1 is one of regular reparameterizations of the circle arc, so by definition of the parametric distance we have dist H c τ, p n dist P c τ, p n dist R c τ, p n. This fact, Lemma 1 and 8 lead to the following observation. Corollary 3 Let p n be the parametric polynomial circle arc approximation defined in 0. Then dist P c τ, p n = dist c τ, p n = a 1 a a.

10 10 Gašper Jaklič, Jernej Kozak 3 Asymptotic analysis Let us examine the circle arc approximation p n properties as degree of the curve n grows. With the expansion arctanu = u u3 3 + u in mind we observe from 19 that the function g approximately equals g s = 1 s n 1 k=0 µ k ν k,n 1 3s 3 n 1 k=0 µ k ν k,n s 5 n 1 k=0 µ k ν k,n = 1 ν 0 s 3ν0 9ν 0 1ν ν0 6 75ν ν0 8 s3 5ν 0 64ν0 6 18ν , 80ν 0 15 s5 where π ν 0 = sin = n π n 1 π O 6 n n 5. If one rewrites the equation 19 in the form s = τ πν 0 τ 3ν0 τ 40ν 6 3πν 0 4ν ν ν0 8 3 s 5πν 0 4ν0 3 16ν , 0ν s4 the fixed point iteration reveals the asymptotic expansion s = τ πν 0 + O 3 ν 0 πν 0 9τ = τ π n + 3τ 4π 36τ 1 1 n + O n 3. From here we observe n ln s s = n ln πν 0 τ + ln 1 π nν0 18τ = n ln n + ln π 4τ n + O 1 n + O ν0. Thus 17 yields a = s + n 1 + s + O s + 6n 1 + s πν0 n = 1 n π ν0 τ 18τ + O ν0 π n ln n + ln = e 4τ n 1 + O π n 1 = 1 + O. 4τ n n 1 n 3

11 On Parametric Polynomial Circle Approximation 11 Finally, the distance between c τ and p n asymptotically equals dist c τ, p n = a + a O a 5 πν0 n = 1 n π ν0 τ π = 4τ n n 1 + O 18τ + O ν0 n+ 1. n 4 4 Numerical considerations 4.1 Computation of s Given n and τ, it takes only a couple of steps of the tangent method to determine s. A very close starting value is provided by the first two terms of. The larger τ 1 Table 1 Values s, γ, and a for τ = 1, i.e., the whole circle approximation. n s asymptot. s #steps γ a Table Values s, γ, and a for τ = 4, i.e., the circle arc [ π 4, π 4 ] approximation. n s asymptot. s #steps γ a is, the fewer iteration steps are required Tables 1 and. Details can be found in [1].

12 1 Gašper Jaklič, Jernej Kozak Note that the equation 19 could be rewritten in a polynomial form too. Suppose n is given, and let us apply tangent to both sides of 19, π tan g s = tan. 5 τ From a well known relation tan arctan u + v = u + tan v 1 u tan v we deduce that the left-hand side of 5 is a rational function of s. By a proper multiplication thus 5 yields its polynomial form ζ n s = 0. Since tan π τ = tan π τ + mπ = 0, m N 0, the polynomial ζ n may have several positive roots, but the largest is s. As an example, the first three polynomials ζ n read ζ 3 s = 8s 1 cos π τ + s 5 4s sin π τ, ζ 4 s = 4 + s 1 s cos π τ + 8s s + 1 sin π τ, ζ 5 s = s 4 s s s s cos π τ, sin π τ. 6 The whole circle case simplifies 6 further, and it is straightforward to determine the corresponding optimal parameters for n = 3, 4, 5, respectively: s = 1, a = 16 65, γ =, s = 1, 1 3/4 a = , γ = , 7 s = , a 3 = , γ = Thus, by 7, the exact distance between the circle and its approximant is available. For example, in the cubic case, the radial error is 1/8 see Table 3.

13 On Parametric Polynomial Circle Approximation Curves p n Curves p n depend on n as well as on τ. So one could divide computations in two steps. In advance, the polynomial curves are computed, depending on the symbolic parameter s that defines γ too. For n = 3, 4, 5, these polynomials are 3 x 3 t = γs 4 + s t, 3 y 3 t = γt 4 + s t, 1 x 4 t = γ 8 + s + s 4 + s + 1 t + t 4, 10 + y 4 t = γst + x 5 t = γs s y 5 t = γt s s t, s s t + 5s s t 4 + 5, 16 t + t 4 and the rest is left to [1]. Once s is determined, so is γ, and the coefficients of p n can be computed in a stable way since they are polynomials in s, with coefficients of the same sign. By Remark 1, note also that a careful evaluation of p nt requires n flops only. From 7 and 8, closed form polynomials for the whole circle case could be deduced for n = 3, 4, 5. Note that for n = 3 and n = 4, closed form polynomials can be obtained for arbitrary circle arc τ 1. The former case requires solving a cubic equation, and the latter just a quadratic one. 8, 4.3 Theoretical and actual distance between the curves Theoretical distance between c τ and p n determined in Corollary 3 and the actual distance start to differ when one approaches the working precision. Table 3 shows the comparison between both distances for the whole circle case, and an estimated error decay rate. Since the asymptotic estimate 4 suggests faster than exponential error decay, we assume that a measured function g n behaves as g n = const e ω n ln n + faster decaying terms as functions of n, const > 0. 9 This gives the error decay rate estimate ω as ω 1 n + 1 lnn + 1 n ln n ln gn+1 The estimated decay rates in Table 3 clearly suggest that n 1 should be used when using the double IEEE precision. One observes also that the assumption made in 9 g n.

14 14 Gašper Jaklič, Jernej Kozak Table 3 The whole circle case: theoretical and actual distance between both curves, with estimated decay rate. n a 1 a a decay actual decay rate radial error rate is sensible, since ω slowly diminishes to as expected from 4. For the degree n = 11, one obtains the polynomial components of the approximation p 11 as x 11 t = t t t t t + 1., y 11 t = t t t t t t, and Fig. 3 and Fig. 4 show the radial error function as well as the difference between the curvature of the approximation and the circle, respectively. Radial error function Fig. 3 The radial error function of the circle approximation p 11.

15 On Parametric Polynomial Circle Approximation 15 Fig. 4 The difference between the curvature of the circle approximation p 11 and the circle. 4.4 Evaluation of trigonometric functions Theoretically, one could evaluate a trigonometric function by, firstly, transforming the given angle θ to a smaller one by using periodicity it suffices to consider θ [ π 4, π 4 ], and then applying Taylor polynomial approximation. However, there are better ways. In practice, CORDIC algorithm is used see [0], e.g.. By using a precomputed table of arctan i, i = 1,,..., and repeated rotations, the values of sin and cos are obtained. In most cases, 40 iterations are sufficient to get the result, correct to the 10th decimal place. Here, we suggest a simple polynomial alternative that could sometimes turn out useful. It is based upon the circle arc approximation derived in the preceding sections. Suppose that one has already reduced the angle interval, and we are looking for an approximation of sin θ and cos θ, θ [ π 4, π 4 ]. Though the parametric polynomial curves p n = x n, yn, computed for the particular circle arc, are its jolly good approximations, so the image of its components x nt, y nt, t [ 1, 1] as a set of points is very close to the c τ, τ = 4. But to obtain an approximation of cos θ or sin θ, a connection between both parameters, t = ψ θ is required. Here, it is straightforward to obtain it. The radial error function 9 vanishes by 3 at parameter values t k, k = 0, 1,..., n 1, very close to the zeroes of the polynomial T n. Lemma 3 Let the radial error function e be given by 9 with 0 < a < 1. Its zeroes are k + 1 t k = cos π k n arcsin a 1 +, k = 0, 1,..., n 1. 1 a 30

16 16 Gašper Jaklič, Jernej Kozak Proof From k T n t k = cos n arccos t k = cos π + 1 k a arcsin a = 1 a a we obtain e t k = and the proof is completed. 1 a a a 1 a a = 0, But at t k given by 30, the circle arc and its approximation coincide. By Theorem then there is precisely one θ k [ π τ, ] π τ that satisfies x n t k cos θk ynt k =, i.e., θ sin θ k = arctan y nt k k x nt k = 0, 1,..., n 1. k, Thus the parameterization t = ψ θ could be simply an interpolating polynomial ψ = ψ n 1 R[θ] of degree n 1 that satisfies the conditions ψ n 1 θ k = t k, k = 0, 1,..., n 1. Table 4 clearly shows that better than exponential approximation order is almost Table 4 Absolute errors in the approximation of cos θ and sin θ, with estimated decay rate. n error in cos θ decay error in sin θ decay approximation rate approximation rate preserved. Also, it indicates that n = 8 is a sensible degree choice when computing cos θ and sin θ approximation in the double IEEE precision. This brings out x 8 = t t t t + 1., y 8 = t t t t, and the parameterization obtained equals ψ 15 θ =1.7673θ θ θ θ θ θ θ θ 15.

17 On Parametric Polynomial Circle Approximation 17 Details can be found in [1]. Note that an evaluation of ψ n 1 θ, x n, and yn requires flops. Already for n = 3, the graphs of the functions and their approximants for θ [ π 4, π 4 ] are indistinguishable see the relative error plot in Fig. 5. Relativeerroratappr.ofCos,Sin, [- π 4,π 4,byp Fig. 5 Relative error at approximation of cos θ blue and sin θ red, θ [ π 4, π ], by using the approximant of degree 4 3. References 1. Ahn, Y.J., Kim, H.O.: Approximation of circular arcs by Bézier curves. J. Comput. Appl. Math. 811, Dokken, T.: Aspects of intersection algorithms and approximation. University of Oslo PhD Thesis 3. Dokken, T.: Controlling the shape of the error in cubic ellipse approximation. In: Curve and surface design Saint-Malo, 00, Mod. Methods Math., pp Nashboro Press, Brentwood, TN Dokken, T., Dæhlen, M., Lyche, T., Mørken, K.: Good approximation of circles by curvaturecontinuous Bézier curves. Comput. Aided Geom. Design 71-4, Curves and surfaces in CAGD 89 Oberwolfach, Fang, L.: Circular arc approximation by quintic polynomial curves. Comput. Aided Geom. Design 158, Fang, L.: G 3 approximation of conic sections by quintic polynomial curves. Comput. Aided Geom. Design 168, Floater, M.: High-order approximation of conic sections by quadratic splines. Comput. Aided Geom. Design 16, Floater, M.S.: An Oh n Hermite approximation for conic sections. Comput. Aided Geom. Design 14, Goldapp, M.: Approximation of circular arcs by cubic polynomials. Comput. Aided Geom. Design 83, Hur, S., Kim, T.: The best G 1 cubic and G quartic Bézier approximations of circular arcs. J. Comput. Appl. Math. 366, Jaklič, G.: Uniform approximation of a circle by a parametric polynomial curve. Comput. Aided Geom. Design 41,

18 18 Gašper Jaklič, Jernej Kozak 1. Jaklič, G., Kozak, J., Krajnc, M., Žagar, E.: On geometric interpolation of circle-like curves. Comput. Aided Geom. Design 45, Jaklič, G., Kozak, J., Krajnc, M., Žagar, E.: Approximation of circular arcs by parametric polynomial curves. Annali dell Universita di Ferrara 53, Jaklič, G., Kozak, J., Krajnc, M., Vitrih, V., Žagar, E.: High-order parametric polynomial approximation of conic sections. Const. Approx. 38 1, Kim, S.H., Ahn, Y.J.: An approximation of circular arcs by quartic Bézier curves. Comput. Aided Design 396, Kovač, B. and Žagar, E.: Some new G 1 quartic parametric approximants of circular arcs. Appl. Math. Comp. 39, Lyche, T., Mørken, K.: A metric for parametric approximation. In: Curves and surfaces in geometric design Chamonix-Mont-Blanc, 1993, pp A K Peters, Wellesley, MA Mørken, K.: Best approximation of circle segments by quadratic Bézier curves. In: Curves and surfaces Chamonix-Mont-Blanc, 1990, pp Academic Press, Boston, MA Piegl, L.A., Tiller, W. and Rajab, K.: It is time to drop the R from NURBS. Eng. with Comput. 30, CORDIC Wikipedia, Accessed February Kozak, J.: On Parametric Polynomial Circle Approximation notebook support kozak/raziskovalnodelo/ NekateriClanki/OnParametricPolynomialCircleApproximation/ programi/onparametricpolynomialcircleapproximation.nb 016. Accessed February 016