On parametric polynomial circle approximation

Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

“…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.…”

“…Since the approximation of a circular arc is considered, one can use the radial distance d rad as the error measure (Degen (1992)). It is a special type of parametric distance considered in Lyche and Mørken (1994) and later in Jaklič and Kozak (2018) where the authors have proved that it actually coincides with the Hausdorff distance in the case of circular arc approximation. Let p = (p x , p y ) T be a PH curve of degree seven approximating the circular arc c given by some small inner angle 2α in the canonical position.…”

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

“…Quintic polynomial approximants of various geometric smoothness have been studied in [3]. The methods how to approximate the whole circle can be found in [6] and in [7]. In the latter paper the Hausdorff distance has been considered but only for the case where interpolation of the boundary points is not required.…”

“…All the above studies involve the simplified radial error (2.3) and it seems that no results are available in the literature on optimal approximation with respect to the the radial error (2.2). However, it is important to study the existence and the uniqueness of the optimal approximants with respect to the radial error, since it was was shown in [7] that the radial error induces the Hausdorff distance in this case. In order to see that things are much more complicated if the radial error is considered, let us take a quick look at the parabolic case first.…”

Optimal parametric interpolants of circular arcs

“…One of the standard measures in this case is the radial distance d r , measuring the distance of the point on the parametric polynomial to the corresponding point on the circular arc in the radial direction. Under some assumptions the metric d r is equivalent to the Hausdorff metric ( [1] and [5]). Hence to find the best interpolant of the unit circular arc c with respect to the Hausdorff metric, we have to find an interpolant p = (x, y) T which minimizes the value d r (c, p) = max t p(t) − 1 = max t…”