2019 **Abstract:** We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the Bézier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a…

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“…But even more important is the analysis of some low degree cases for which n − k > 2. The first step in this direction was done in [16] and in [15] but only for the simplified radial error. It is clear that the problem of finding the best geometric interpolant in the case of the radial error is much more challenging issue.…”

confidence: 99%

“…In the latter paper the Hausdorff distance has been considered but only for the case where interpolation of the boundary points is not required. A general framework for the approximation of circular arcs by parametric polynomials was given in [16] and some particular cases following this approach can be found in [15]. The optimal approximants of maximal geometric smoothness are characterized in [9].…”

confidence: 99%

“…Most of the approximation techniques namely consider the interpolation of local geometric data only and use the remaining parameters to minimize the distance between the interpolant and the circular arc, to minimize the deviation of the curvature, etc. The results of this type can be found in Dokken et al (1990), Goldapp (1991), Lyche and Mørken (1994), Mørken (1995), Ahn and Kim (1997), Kim and Ahn (2007), Jaklič et al (2007), Jaklič et al (2013), Kovač and Žagar (2016), Jaklič (2016), Jaklič and Kozak (2018), Knez and Žagar (2018), Vavpetič and Žagar (2019), Ahn (2019), Vavpetič (2020) and Vavpetič and Žagar (2021), if we mention just the most important and recent ones. Although the proposed algorithms provide good approximations of circular arcs if the Hausdorff distance is considered as a measure of the error, they do not include an arc length in interpolation data.…”

confidence: 85%

“…This is due to the fact that a rotational cylinder cannot carry a polynomial curve transversal to the rulings as it would project onto a circle. While a circle does not have an exact polynomial parameterization, it is possible to achieve good approximations with cubics (see [33] and the references therein). This is sufficient for our purposes.…”

confidence: 99%