A class of generalized quadratic B-splines with local controlling functions

Qi Xie; Yiting Huang; Qi Xie; Yiting Huang

doi:10.3934/math.20231193

AIMS Mathematics

2023, Volume 8, Issue 10: 23472-23499. doi: 10.3934/math.20231193

Previous Article Next Article

Research article

A class of generalized quadratic B-splines with local controlling functions

Qi Xie ,
Yiting Huang ^,

School of Mathematics, South China University of Technology, Guangzhou 510641, China

Received: 27 May 2023 Revised: 19 July 2023 Accepted: 24 July 2023 Published: 27 July 2023
MSC : 41A15

In this work, a class of generalized quadratic Bernstein-like functions having controlling functions is constructed. It contains many particular cases from earlier papers. Regarding the controlling functions, sufficient conditions are given. Corner cutting algorithms and the accompanying quadratic Bézier curves are discussed. A class of generalized quadratic B-splines possessing controlling functions is proposed. Some important properties for curve and surface design are proved. Sufficient conditions for $ C^2 $ continuity, $ C^3 $ continuity and $ C^n $ continuity are also given. Some applications of the constructed B-splines in $ \mathbb{R}^2 $ and $ \mathbb{R}^3 $ are presented, which show the ability to adjust the shape of the curves flexibly and locally. These applications show that generalized quadratic B-splines can be easily implemented and serve as an alternative strategy for modeling curves.
- quadratic Bernstein-like functions,
- quadratic B-splines,
- extended Chebyshev spaces,
- $ C^n $ continuity,
- curve design
Citation: Qi Xie, Yiting Huang. A class of generalized quadratic B-splines with local controlling functions[J]. AIMS Mathematics, 2023, 8(10): 23472-23499. doi: 10.3934/math.20231193

Related Papers:

Abstract

In this work, a class of generalized quadratic Bernstein-like functions having controlling functions is constructed. It contains many particular cases from earlier papers. Regarding the controlling functions, sufficient conditions are given. Corner cutting algorithms and the accompanying quadratic Bézier curves are discussed. A class of generalized quadratic B-splines possessing controlling functions is proposed. Some important properties for curve and surface design are proved. Sufficient conditions for $ C^2 $ continuity, $ C^3 $ continuity and $ C^n $ continuity are also given. Some applications of the constructed B-splines in $ \mathbb{R}^2 $ and $ \mathbb{R}^3 $ are presented, which show the ability to adjust the shape of the curves flexibly and locally. These applications show that generalized quadratic B-splines can be easily implemented and serve as an alternative strategy for modeling curves.

References

[1]	R. Wang, C. Li, C. Zhu, Computational geometry tutorial (Chinese), Beijing: Science Press, 2008.
[2]	X. Zhu, Free-form curve and surface modeling technology (Chinese), Beijing: Science Press, 2000.
[3]	R. Wang, Numerical rational approximation (Chinese), Shanghai: Shanghai Scientific and Technical Publishers, 1980.
[4]	C. de Boor, On calculating with B-splines, J. Approx. Theory, 6 (1972), 50–62. https://doi.org/10.1016/0021-9045(72)90080-9 doi: 10.1016/0021-9045(72)90080-9
[5]	K. Versprille, Computer-aided design applications of the rational b-spline approximation form, Ph. D Thesis, Syracuse University, 1975.
[6]	L. Ramshaw, Blossoming: a connect-the-dots approach to splines, Palo Alto: Digital Equipment Corporation, 1987.
[7]	M. Mazure, P. Laurent, Piecewise smooth spaces in duality: application to blossoming, J. Approx. Theory, 98 (1999), 316–353. https://doi.org/10.1006/jath.1998.3306 doi: 10.1006/jath.1998.3306
[8]	M. Mazure, Quasi-chebyshev splines with connection matrices: application to variable degree polynomial splines, Comput. Aided Geom. D., 18 (2001), 287–298. https://doi.org/10.1016/s0167-8396(01)00031-0 doi: 10.1016/s0167-8396(01)00031-0
[9]	M. Mazure, Blossoms and optimal bases, Adv. Comput. Math., 20 (2004), 177–203. https://doi.org/10.1023/A: 1025855123163
[10]	P. Costantini, T. Lyche, C. Manni, On a class of weak Tchebycheff systems, Numer. Math., 101 (2005), 333–354. https://doi.org/10.1007/s00211-005-0613-6 doi: 10.1007/s00211-005-0613-6
[11]	M. Mazure, On dimension elevation in quasi extended Chebyshev spaces, Numer. Math., 109 (2008), 459–475. https://doi.org/10.1007/s00211-007-0133-7 doi: 10.1007/s00211-007-0133-7
[12]	P. Costantini, F. Pelosi, M. Sampoli, New spline spaces with generalized tension properties, BIT Numer. Math., 48 (2008), 665–688. https://doi.org/10.1007/s10543-008-0195-7 doi: 10.1007/s10543-008-0195-7
[13]	X. Han, S. Liu, Extension of a quadratic Bézier curve (Chinese), Journal of Central South University (Science and Technology), 34 (2003), 214–217.
[14]	J. Xie, S. Hong, Quadratic B-spline curve with shape parameters (Chinese), Computer Aided Engineering, 15 (2006), 15–19.
[15]	L. Yan, T. Liang, The quadratic Bézier curves that shape can adjust (Chinese), Journal of East China University of Technology (Natural Science), 31 (2008), 93–97.
[16]	X. Han, Quadratic trigonometric polynomial curves with a shape parameter, Comput. Aided Geom. D., 19 (2002), 503–512. https://doi.org/10.1016/s0167-8396(02)00126-7 doi: 10.1016/s0167-8396(02)00126-7
[17]	X. Wu, Research on the theories and methods of geometric modeling based on the curves and surfaces with shape parameter, Ph. D Thesis, Central South University, 2008.
[18]	X. Han, Quadratic trigonometric polynomial curves concerning local control, Appl. Numer. Math., 56 (2006), 105–115. https://doi.org/10.1016/j.apnum.2005.02.013 doi: 10.1016/j.apnum.2005.02.013
[19]	X. Han, Piecewise quadratic trigonometric polynomial curves, Math. Comput., 72 (2003), 1369–1377. https://doi.org/10.1090/s0025-5718-03-01530-8 doi: 10.1090/s0025-5718-03-01530-8
[20]	X. Han, $C^2$ quadratic trigonometric polynomial curves with local bias, J. Comput. Appl. Math., 180 (2005), 161–172. https://doi.org/10.1016/j.cam.2004.10.008 doi: 10.1016/j.cam.2004.10.008
[21]	U. Bashir, M. Abbas, J. Ali, The $G^{2}$ and $C^{2}$ rational quadratic trigonometric Bézier curve with two shape parameters with applications, Appl. Math. Comput., 219 (2013), 10183–10197. https://doi.org/10.1016/j.amc.2013.03.110 doi: 10.1016/j.amc.2013.03.110
[22]	L. Yan, X. Han, Q. Zhou, Quadratic hyperbolic Bézier curve and surface (Chinese), Computer Engineering and Science, 37 (2015), 162–167.
[23]	F. Pelosi, R. Farouki, C. Manni, A. Sestini, Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics, Adv. Comput. Math., 22 (2005), 325–352. https://doi.org/10.1007/s10444-003-2599-x doi: 10.1007/s10444-003-2599-x
[24]	R. Ait-Haddou, M. Barton, Constrained multi-degree reduction with respect to Jacobi norms, Comput. Aided Geom. D., 42 (2016), 23–30. https://doi.org/10.1016/j.cagd.2015.12.003 doi: 10.1016/j.cagd.2015.12.003
[25]	C. González, G. Albrecht, M. Paluszny, M. Lentini, Design of $ C^2 $ algebraic-trigonometric Pythagorean hodograph splines with shape parameters, Comput. Appl. Math., 37 (2018), 1472–1495. https://doi.org/10.1007/s40314-016-0404-y doi: 10.1007/s40314-016-0404-y
[26]	M. Mazure, Which spaces for design? Numer. Math., 110 (2008), 357–392. https://doi.org/10.1007/s00211-008-0164-8
[27]	M. Mazure, On a general new class of quasi Chebyshevian splines, Numer. Algor., 58 (2011), 399–438. https://doi.org/10.1007/s11075-011-9461-x doi: 10.1007/s11075-011-9461-x
[28]	M. Mazure, Quasi extended Chebyshev spaces and weight functions, Numer. Math., 118 (2011), 79–108. https://doi.org/10.1007/s00211-010-0312-9 doi: 10.1007/s00211-010-0312-9
[29]	T. Bosner, M. Rogin, Variable degree polynomial splines are Chebyshev splines, Adv. Comput. Math., 38 (2013), 383–400. https://doi.org/10.1007/s10444-011-9242-z doi: 10.1007/s10444-011-9242-z
[30]	L. Schumaker, Spline functions: basic theory, Cambridge: Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511618994
[31]	H. Pottmann, The geometry of Tchebycheffian splines, Comput. Aided Geom. D., 10 (1993), 181–210. https://doi.org/10.1016/0167-8396(93)90036-3 doi: 10.1016/0167-8396(93)90036-3
[32]	M. Mazure, Blossoming: a geometrical approach, Constr. Approx., 15 (1999), 33–68. https://doi.org/10.1007/s003659900096 doi: 10.1007/s003659900096
[33]	M. Mazure, Blossoming stories, Numer. Algor., 39 (2005), 257–288. https://doi.org/10.1007/s11075-004-3642-9
[34]	M. Mazure, On a new criterion to decide whether a spline space can be used for design, BIT Numer. Math., 52 (2012), 1009–1034. https://doi.org/10.1007/s10543-012-0390-4 doi: 10.1007/s10543-012-0390-4
[35]	J. Peña, Shape preserving representations in computer-aided geometric design, New York: Nova Science Publishers, 1999.
[36]	G. Farin, Curves and surfaces for computer-aided geometric design: a practical guide, Boston: Academic Press, 1990. https://doi.org/10.1016/C2009-0-22351-8
[37]	J. Peña, Shape preserving representations for trigonometric polynomial curves, Comput. Aided Geom. D., 14 (1997), 5–11. https://doi.org/10.1016/s0167-8396(96)00017-9 doi: 10.1016/s0167-8396(96)00017-9
[38]	J. Carnicer, J. Peña, Totally positive bases for shape preserving curve design and optimality of B-splines, Comput. Aided Geom. D., 11 (1994), 633–654. https://doi.org/10.1016/0167-8396(94)90056-6 doi: 10.1016/0167-8396(94)90056-6
[39]	L. Gori, F. Pitolli, Totally positive refinable functions with general dilation, Appl. Numer. Math., 112 (2017), 17–26. https://doi.org/10.1016/j.apnum.2016.10.004 doi: 10.1016/j.apnum.2016.10.004

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)