On \(G^2\) approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines. (English) Zbl 07923119
Summary: The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining \(G^2\) smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all \(x\)-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the \(G^2\) interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 41A15 | Spline approximation |
| 14H50 | Plane and space curves |
References:
| [1] | Berberich, Eric; Emeliyanenko, Pavel; Kobel, Alexander; Sagraloff, Michael, Exact symbolic-numeric computation of planar algebraic curves, Theor. Comput. Sci., 491, 1-32, 2013 · Zbl 1277.68300 |
| [2] | Cheng, Jin-San; Jin, Kai, A generic position based method for real root isolation of zero-dimensional polynomial systems, J. Symb. Comput., 68, 204-224, 2015 · Zbl 1304.13048 |
| [3] | Cheng, Jin-San; Gao, Xiao-Shan; Guo, Leilei, Root isolation of zero-dimensional polynomial systems with linear univariate representation, International Symposium on Symbolic and Algebraic Computation (ISSAC 2009). International Symposium on Symbolic and Algebraic Computation (ISSAC 2009), J. Symb. Comput., 47, 7, 843-858, 2012 · Zbl 1254.65064 |
| [4] | Farouki, Rida T., The elastic bending energy of Pythagorean-hodograph curves, Comput. Aided Geom. Des., 13, 3, 227-241, 1996 · Zbl 0875.68861 |
| [5] | Farouki, Rida T., Pythagorean-Hodograph Curves, 2008, Springer · Zbl 1144.51004 |
| [6] | Farouki, Rida T.; Neff, C. Andrew, Hermite interpolation by Pythagorean hodograph quintics, Math. Comput., 64, 212, 1589-1609, 1995 · Zbl 0847.68125 |
| [7] | Farouki, Rida T.; Sakkalis, Takis, Pythagorean hodographs, IBM J. Res. Dev., 34, 5, 736-752, 1990 |
| [8] | Farouki, Rida T.; Hormann, Kai; Nudo, Federico, Singular cases of planar and spatial \(C^1\) Hermite interpolation problems based on quintic Pythagorean-hodograph curves, Comput. Aided Geom. Des., 82, Article 101930 pp., 2020 · Zbl 1450.65006 |
| [9] | Farouki, Rida T.; Pelosi, Francesca; Sampoli, Maria Lucia, Approximation of monotone clothoid segments by degree 7 Pythagorean-hodograph curves, J. Comput. Appl. Math., 382, Article 113110 pp., 2021 · Zbl 1484.65035 |
| [10] | Gao, Xiao-Shan; Li, Ming, Rational quadratic approximation to real algebraic curves, Geometric Modeling and Processing 2004. Geometric Modeling and Processing 2004, Comput. Aided Geom. Des., 21, 8, 805-828, 2004 · Zbl 1069.65518 |
| [11] | Gonzalez-Vega, Laureano; Necula, Ioana, Efficient topology determination of implicitly defined algebraic plane curves, Comput. Aided Geom. Des., 19, 9, 719-743, 2002 · Zbl 1043.68105 |
| [12] | He, Shitao; Shen, Li-Yong; Wu, Qin; Yuan, Chunming, A certified cubic b-spline interpolation method with tangential direction constraints, J. Syst. Sci. Complex., 37, 3, 1271-1294, 2024 · Zbl 07903357 |
| [13] | Jaklič, Gašper; Kozak, Jernej; Krajnc, Marjeta; Vitrih, Vito; Žagar, Emil, On interpolation by planar cubic \(G^2\) Pythagorean-hodograph spline curves, Math. Comput., 79, 269, 305-326, 2010 · Zbl 1200.41003 |
| [14] | Jaklič, Gašper; Kozak, Jernej; Krajnc, Marjeta; Vitrih, Vito; Žagar, Emil, Interpolation by \(G^2\) quintic Pythagorean-hodograph curves in \(\mathbb{R}^d\), Numer. Math., Theory Methods Appl., 7, 3, 374-398, 2014 · Zbl 1324.65011 |
| [15] | Jin, Kai; Cheng, Jinsan, On the topology and isotopic meshing of plane algebraic curves, J. Syst. Sci. Complex., 33, 1, 230-260, 2020 · Zbl 1439.14106 |
| [16] | Jüttler, Bert, Hermite interpolation by Pythagorean hodograph curves of degree seven, Math. Comput., 70, 235, 1089-1111, 2001 · Zbl 0963.68210 |
| [17] | Knez, Marjeta; Pelosi, Francesca; Sampoli, Maria Lucia, Construction of \(G^2\) planar Hermite interpolants with prescribed arc lengths, Appl. Math. Comput., 426, Article 127092 pp., 2022 · Zbl 1510.65038 |
| [18] | Pérez-Díaz, Sonia; Sendra, Juana; Sendra, J. Rafael, Parametrization of approximate algebraic curves by lines, Algebraic and Numerical Algorithms. Algebraic and Numerical Algorithms, Theor. Comput. Sci., 315, 2, 627-650, 2004 · Zbl 1085.68187 |
| [19] | Shen, Li-Yong; Yuan, Chun-Ming; Gao, Xiao-Shan, Certified approximation of parametric space curves with cubic B-spline curves, Comput. Aided Geom. Des., 29, 8, 648-663, 2012 · Zbl 1251.65014 |
| [20] | Shou, Huahao; Shi, Wen; Miao, Yongwei, Biarc approximation of planar algebraic curve, (Zhou, Qihai, Theoretical and Mathematical Foundations of Computer Science. Theoretical and Mathematical Foundations of Computer Science, Berlin, Heidelberg, 2011, Springer: Springer Berlin, Heidelberg), 456-463 |
| [21] | Wang, Xin-Yu; Shen, Li-Yong; Yuan, Chun-Ming; Pérez-Díaz, Sonia, Globally certified \(G^1\) approximation of planar algebraic curves, J. Comput. Appl. Math., 436, Article 115399 pp., 2024 · Zbl 07738649 |
| [22] | Zeng, Guangxing, Determination of the tangents for a real plane algebraic curve, J. Symb. Comput., 41, 8, 863-886, 2006 · Zbl 1121.14046 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
