Approximation of conic sections by weighted Lupaş post-quantum Bézier curves

• Autorzy: Khan Asif , Iliyas Mohammad , Khan Khalid , Mursaleen Mohammad

Identyfikatory

DOI

10.1515/dema-2022-0016

• Języki publikacji: EN

Abstrakty

EN

This paper deals with weighted Lupaş post-quantum Bernstein blending functions and Bézier curves constructed with the help of bases via (p,q) -integers. These blending functions form normalized totally positive bases. Due to the rational nature of weighted Lupaş post-quantum Bézier curves and positive weights, they help in investigating from geometric point of view. Their degree elevation properties and de Casteljau algorithm have been studied. It has been shown that quadratic weighted Lupaş post-quantum Bézier curves can represent conic sections in two-dimensional plane. Graphical analysis has been presented to discuss geometric interpretation of weight and conic section representation by weighted Lupaş post-quantum Bézier curves. This new generalized weighted Lupaş post-quantum Bézier curve provides better approximation and flexibility to a particular control point as well as control polygon due to extra parameter p and q in comparison to classical rational Bézier curves, Lupaş q -Bézier curves and weighted Lupaş q -Bézier curves.

Słowa kluczowe

EN

post-quantum integers; weighted Lupaş post-quantum Bézier curve; rational Bezier curves; normalized totally positive basis; shape parameter; conic section

PL

liczby całkowite; krzywa Béziera; parametr kształtu; sekcja stożkowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 328--342
• Opis fizyczny: Bibliogr. 36 poz., wykr.

Twórcy

autor

Khan Asif

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

autor

Iliyas Mohammad

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

autor

Khan Khalid

• School of Computer and System Sciences, SC & SS, J.N.U. New Delhi 110067, India

autor

Mursaleen Mohammad

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
• Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

Bibliografia

• [1] S. N. Bernstein, Constructive proof of Weierstrass approximation theorem, Commun. Soc. Math. Kharkov 13 (1912), 1–2.
• [2] T. N. T. Goodman, Shape preserving representations, in: T. Lyche and L. L. Shumaker (eds), Mathematical Methods in CAGD, Academic Press, Boston, 1989, pp. 333–357.
• [3] J. M. Carnicer and J. M. Peña, Totally positive bases for shape preserving curve design and optimality of B-splines, Comput. Aided Geom. Design 11 (1994), no. 6, 633–654, DOI: https://doi.org/10.1016/0167-8396(94)90056-6.
• [4] X. L. Han, Normalized B-basis of the space of trigonometric polynomials and curve design, Appl. Math. Comput. 251 (2015), 336–348, DOI: https://doi.org/10.1016/j.amc.2014.11.070.
• [5] E. Mainar and J. M. Peña, Optimal bases for a class of mixed spaces and their associated spline spaces, Comput. Math. Appl. 59 (2010), 1509–1523.
• [6] L.-W. Han, Y. Chu, and Z.-Y. Qiu, Generalized Bézier curves and surfaces based on Lupaş q-analogue of Bernstein operator, J. Comput. Appl. Math. 261 (2014), 352–363, DOI: https://doi.org/10.1016/j.cam.2013.11.016.
• [7] H. Oruç and G. M. Phillips, q-Bernstein polynomials and Bézier curves, J. Comput. Appl. Math. 151 (2003), 1–12, DOI: https://doi.org/10.1016/S0377-0427(02)00733-1.
• [8] Kh. Khan and D. K. Lobiyal, Bézier curves based on Lupaş (p,q)-analogue of Bernstein functions in CAGD, J. Comput. Appl. Math. 317 (2017), 458–477, DOI: https://doi.org/10.1016/j.cam.2016.12.016.
• [9] K. Victor and C. Pokman, Quantum Calculus, Springer-Verlag, New York Berlin Heidelberg, 2002.
• [10] A. Lupaş, A q-analogue of the Bernstein operator, in: Seminar on Numerical and Statistical Calculus, vol. 9, University of Cluj-Napoca, 1987, p. 85–92.
• [11] G. M. Phillips, Bernstein polynomials based on the q-integers, The heritage of P. L. Chebyshev, Ann. Numer. Math. 4 (1997), 511–518.
• [12] L.-W. Han, Y.-S. Wu, and Y. Chu, Weighted Lupaş q-Bézier curve, J. Comput. Appl. Math. 308 (2016), 318–329, DOI: https://doi.org/10.1016/j.cam.2016.06.017.
• [13] M. Mursaleen, K. J. Ansari, and A. Khan, On (p q, )-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882, [Erratum: 278 (2016), 70-71], DOI: https://doi.org/10.1016/j.amc.2016.02.008.
• [14] M. Mursaleen, F. Khan, and A. Khan, Approximation by (p q, )-Lorentz polynomials on a compact disk, Complex Anal. Oper. Theory 10 (2015), 1725–1740.
• [15] T. Acar, A. Aral, and M. Mursaleen, Approximation by Baskakov-Durrmeyer operators based on (p q, )-integers, Math. Slovaca 68 (2018), no. 4, 897–906, DOI: https://doi.org/10.1515/ms-2017-0153.
• [16] T. Acar, S. A. Mohiuddine and M. Mursaleen, Approximation by (p q, )-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory 12 (2018), no. 6, 1453–1468, DOI: https://doi.org/10.1007/s11785-016-0633-5.
• [17] T. Acar, A. Aral, and S. A. Mohiuddine, Approximation by bivariate (p q, )-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 2, 655–662.
• [18] T. Acar, A. Aral, and S. A. Mohiuddine, On Kantorovich modification of (p q, )-Bernstein operators, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 3, 1459–1464.
• [19] S. A. A. Karim, A. Saaban, V. Skala, A. Ghaffar, K. S. Nisar, and D. Baleanu, Construction of new cubic Bézier-like triangular patches with application in scattered data interpolation, Adv. Differ. Equ. 2020 (2020), 151, DOI: https://doi.org/10.1186/s13662-020-02598-w.
• [20] A. Kajla, S. A. Mohiuddine, and A. Alotaibi, Blending-type approximation by Lupaş-Durrmeyer-type operators involving Polya distribution, Math. Methods Appl. Sci. 44 (2021), no. 11, 9407–9418, DOI: https://doi.org/10.1002/mma.7368.
• [21] M. L. Mazure, Extended Chebyshev spaces in rationality, BIT 53 (2013), 1013–1045, DOI: https://doi.org/10.1007/s10543-013-0432-6.
• [22] N. Rao and A. Wafi, (p q, )-bivariate-Bernstein-Chlodowsky operators, Filomat 32 (2018), no. 2, 369–378, DOI: https://doi.org/10.2298/FIL1802369R.
• [23] A. Rababah and S. Manna, Iterative process for G2-multi degree reduction of Bézier curves, Appl. Math. Comput. 217 (2011), 8126–8133.
• [24] G. M. Phillips, A de Casteljau algorithm for generalized Bernstein polynomials, BIT 36 (1996), 232–236.
• [25] J. M. Carnicer, M. Garcìa-Esnaola, and J. M. Peña, Convexity of rational curves and total positivity, J. Comput. Appl. Math. 71 (1996), 365–382.
• [26] M. Mursaleen, Md. Nasiruzzaman, Asif Khan, and K. J. Ansari, Some approximation results on Bleimann-Butzer-Hahn operators defined by post-quantum-integers, Filomat 30 (2016), no. 3, 639–648, DOI: https://doi.org/10.2298/FIL1603639M.
• [27] Q. B. Cai and W. T. Cheng, Convergence of λ-Bernstein operators based on (p q, )-integers, J. Inequal. Appl. 2020 (2020), 35, DOI: https://doi.org/10.1186/s13660-020-2309-y.
• [28] M. N. Hounkonnou and J. Désiré Bukweli Kyemba, p q,( )-calculus: differentiation and integration, SUT J. Math. 49 (2013), no. 2, 145–167.
• [29] M. Gasca and C. A. Micchelli, Total Positivity and its Applications, Kluwer Academic Publishers, Dordrecht, Boston, 1996.
• [30] S. Karlin, Total Positivity, Stanford University Press, London, 1968.
• [31] M. Gasca and J. M. Peña, Total positivity and Neville elimination, Linear Algebra Appl. 165 (1992), 25–44, DOI: https://doi.org/10.1016/0024-3795(92)90226-Z.
• [32] M. Gasca and J. M. Peña, Total positivity, QR factorization and Neville elimination, SIAM J. Matrix Anal. Appl. 14 (1993), 1132–1140.
• [33] T. N. T Goodman, Total positivity and shape of curves, in: M. Gasca and C. A. Miccheli (eds), Total Positivity and its Applications, Kluwer Academic Publishers, Dordrecht, 1996, pp. 157–186.
• [34] L. Piegl, A geometric investigation of rational Bézier scheme of computer aided design, Comput. Ind. Eng. 7 (1986), 401–410, DOI: https://doi.org/10.1016/0166-3615(86)90088-6.
• [35] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Springer-Verlag, Berlin Heidelberg Germany, 1997.
• [36] G. Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th ed., Academic Press, San Diego, 2002.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).