Bézier curves and surfaces are the most widely used mathematical modelling tools in CAD/CAM systems, see [1–3]. One of the main concerns in representing Bézier curves is to keep the degree as low as possible. This simplifies the evaluation, manipulation and determination of a small number of Bézier points. These and other factors encourage us to consider approximating circular arcs using quadratic Bézier curves. Besides many other applications, quadratic Bézier curves are commonly used in encoding and rendering of type fonts and HTML techniques by many companies. Circular arcs are commonly used in the fields of Computer Aided Geometric Design CAGD, Computer Graphics, and many other applications. Since circular arcs are represented by rational Bézier curves and cannot be represented by polynomial curves in explicit form, circular arc representations using polynomial Bézier curves have been developed by many researchers, see for example [4–14].

In this paper, a novel approach to represent a circular arc using quadratic Bézier curves with high accuracy is proposed. The method leads to the solution that minimizes a variation of the Euclidean error.

We want to represent the longest arc of the circle, i.e. the angle *θ* as large as possible. At the same time, the resulting Bézier curve has to satisfy the Chebyshev error. It is known that the angle *θ* can not be greater than $\frac{\pi}{2}$. So, we consider the circular arc *c* : *t* ↦ (cos(*t*), sin(*t*)), −*θ* ≤ *t* ≤ *θ*, where $\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Later, we will find out the largest value for *θ* that satisfies the Chebyshev error. An illustrative choice for the Bézier points with $\theta \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\pi}{4}$ is shown in Fig. 1.

Fig. 1 A circular arc (quarter, $\theta \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\pi}{4}$)

It is not possible to exactly represent a circle with a polynomial curve. While a circle can be represented exactly using rational Bézier curves, a polynomial approximation is preferred in many applications. The ability to represent a primitive circle is a must, especially in computer graphics and data and image processing. Thus, there is a demand to find a parametrically defined polynomial curve *p* : *t* ↦(*x*(*t*), *y*(*t*)), 0 ≤ *t* ≤ 1, where *x*(*t*), *y*(*t*) are polynomials of degree *n*. The degree of *p* has to be as small as possible, and *p* has to approximate *c* within tolerable error. Having the degree *n* low makes the software very fast, convenient, obviates complications of high degree, and reduces the cost. In this paper, degree 2 curves are considered, and it is shown that it works well and produces results that are as good as the results of higher degrees. This makes the method competitive. Namely, quadratic Bézier curves are constructed to represent circular arcs with the best quadratic uniform approximation and the highest accuracy.

A possible function to measure the error between p and c is the Euclidean error function:
$$E(t):=\sqrt{{x}^{2}(t)+{y}^{2}(t)-1.}$$(1)

The square root complicates the analysis. Thus to avoid radicals, we find the square of the *p* components of the circular arc. So, *E*(*t*), is replaced by the following error function
$$e(t):={x}^{2}(t)+{y}^{2}(t)-1.$$(2)

Note that both *e*(*t*), and *E*(*t*), attain their roots and extrema at the same parameters. In this paper, we are interested in finding the quadratic best uniform approximation that has the highest order of approximation and the minimum error. This research is motivated by the conjecture in [11] which states that it is possible to approximate a curve by a polynomial of degree *n* with order 2*n*, rather than the classical order *n* + 1. In quadratic case, the associated error function has to equioscillate five times. Consequently, the approximation problem can be formulated as follows.

*The approximation problem* in this paper is to find *p* : *t* ↦ *x*(*t*), *y*(*t*)), 0 ≤ *t* ≤ 1, where *x*(*t*), *y*(*t*), are polynomials of degree 2, that approximates *c* by satisfying the following three conditions:

*p* minimizes max_{t∈[0;1]} |*e*(*t*)|,

*p* approximates *c* with order four,

*e*(*t*), equioscillates five times over [0, 1].

The solution to this problem is shown in Section 3 to be as follows:
$$x(t)=\left(\frac{3}{2\sqrt{2}}-1\right)+\text{\hspace{0.17em}}4(t-{t}^{2}),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y(t)=\sqrt{\frac{3}{\sqrt{2}}-1}\text{\hspace{0.17em}}(2t-1),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in [0,\text{\hspace{0.17em}}1].$$

It represents the largest circular arc that can satisfy the Chebyshev error. This solution covers almost half of the circle and is presented in Fig. 3; the corresponding error is shown in Fig. 4.

Fig. 3. The circular arc and the quadratic approximating Bézier curve

Fig. 4. Euclidean error of the quadratic approximating Bézier curve

This paper is organized as follows. Section 1 introduces some preliminaries and defines the Bézier points for the best solution (the Bézier curve). The main result is given in Theorem 3.1 in Section 3. In Section 4, the properties of the best solution are presented. Section 5 states all other possible solutions. Section 6 presents comparisons between the quarter of the circle using this method and other existing methods. Conclusions and suggested open problems are given in Section 7.

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