High-Order Parametric Polynomial Approximation of Conic Sections

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.…”

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

“…(3 − 4(λ + 1)a + 6λa 2 )), (17) where λ is given by (14) and a by (16). Two positive zeros t 1,2 of p * 8,1 can be found as a solution of the quadratic equation arising from the quartic factor in (17).…”

“…Let t 1,2 ∈ (0, 1) be positive zeros of the polynomial p * 8,1 , defined by (17). Note that this two zeros can be given in a closed form, since we only have to solve a quadratic equation with exactly known coefficients determined by (17). The nonlinear system for the unknowns ξ and d is given by (19) as…”

“…In [15], the Taylor type geometric interpolation, i.e., interpolation at just one point was considered for all odd degree polynomials. For even degree ones the results can be found in [16] and in a more general form in [17]. The approximation of the whole circle by Lagrange type approximants can be found in [18] and in [4].…”

Approximation of planar curves

“…A lot of work has been done in the past few years in this regard. Some noticeable contributions include [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The existing approximation schemes have focused on the approximation of rational quadratic Bézier curves, which represent conic sections.…”