Computer Science

Computer Science ›› 2023, Vol. 50 ›› Issue (7): 119-128.doi: 10.11896/jsjkx.220800024

• Computer Graphics & Multimedia • Previous Articles     Next Articles

Constructing Combined Quadratic h-Bezier Curves with Monotone Curvature

LI Lin1, XIE Bin2, HAN Liwen1,3,4   

  1. 1 School of Mathematics Sciences,Hebei Normal University,Shijiazhuang 050024,China
    2 College of Computer and Cyber Security,Hebei Normal University,Shijiazhuang 050024,China
    3 Hebei Key Laboratory of Computational Mathematics and Applications,Shijiazhuang 050024,China
    4 Hebei International Joint Research Center for Mathematics and Interdisciplinary Science,Shijiazhuang 050024,China
  • Received:2022-08-01 Revised:2022-11-16 Online:2023-07-15 Published:2023-07-05
  • About author:LI Lin,born in 1999,postgraduate.Her main research interest is computer aided geometric design.HAN Liwen,born in 1974,Ph.D,professor.Her main research interests include computer aided geometric design and computer geometry.
  • Supported by:
    National Natural Science Foundation of China(62076088),National Natural Science Foundation of Hebei Province,China(A2018205103) and Research Fund of Hebei Normal University(L2020Z02).

PDF (PC)

550

Abstract: The h-Bézier curves(h>0),also known as Pólya curves,share many excellent properties with classical Bézier curves(h=0).In this paper,the necessary and sufficient conditions for quadratic h-Bézier curves with monotonic curvature and its construction algorithm are studied.First,the sufficient and necessary conditions for quadratic h-Bézier curves with monotone curvature are obtained by discussing the existence of the extremes of the curves.By introducing curvature critical circles of the curvature,the monotony for the curvature of the quadratic h-Bézier curve can be directly verified by checking whether the middle control point of the curve is on or inside the curvature critical circle.Two algorithms for construct quadratic h-Bezier curves with monotonic curvature are obtained,ensuring that the curves can have monotonically decreasing or increasing curvature by adjusting the shape parameter h.Secondly,the G2 smooth blending of two quadratic h-Bézier curves is studied.Based on the analysis of the properties of the quadratic h-Bézier curves,the shoulder point of the second curve is selected to join with the end point of the first curve,and the necessary and sufficient conditions are obtained and the influence of parameters on the shape of the blending curve is discussed.Finally,the combined quadratic h-Bézier curve with decreasing(or increasing) is constructed.The numerical examples show the modeling advantage and flexibility of the combined quadratic h-Bézier curve.

Key words: h-Bézier curve, Monotone curvature, Curvature critical circles, G2 blending, Shoulder point

CLC Number: 

  • TP391
[1]STANCU D.Approximation of functions by a new class of liner Polynomial operators[J].Revue Roumaine de Mathematiques Pures et Appliquees,1968,13:1173-1194.
[2]GOLDMAN R.Pólya’s urn model and computer aided geometric design[J].SIAM Journal on Algebraic and Discrete Methods,1985,6(1):1-28.
[3]GOLDMAN R.Urn models,approximations,and splines[J].Journal of Approximation Theory,1988,54(1):1-66.
[4]SIMEONOV P,ZAFIRIS V,GOLDMAN R.h-Blossoming:anew approach toalgorithms and identities for h-Bernstein bases and h-Bézier curves Computer Aided Geometric Design,2011,28(9):549-565.
[5]SUN Y H,HAN L W.The rational h-Béziercurve and its representation of conicsection [J].Journal of Computer-Aided Design &Computer Graphics,2019,31(9):1581-1590.
[6]MARCO A,MARTINEA J J,VIAA R.Accurate bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices and applications[J].Linear Algebra and its Applications,2019,579:320-335.
[7]PHILLIPS G M.A de Casteljau algorithm for generalized Bernstein polynomials[J].Bit Numerical Mathematics,1996,36(1):232-236.
[8]ORUÇ H.Generalized Bernstein polynomials and total positivity[D].Scotland:University of St.Andrews,1998.
[9]ORUÇ H,PHILLIPS G M.q-Bernstein polynomials and Bézier curves[J].Journal of Computational and Applied Mathematics,2003,151(1):1-12.
[10]DISIBUYUK C,ORUÇ H.A generalized of rational Bernstein Bézier curves [J].BIT Numerical Mathematics,2007,47:313-323.
[11]SIMEONOV P,ZAFIRIS V,GOLDMAN R.q-Blossoming:A new approach to algorithms and identities for q-Bernstein bases and q-Bézier curves [J].Journal of Approximation Theory,2012,164:77-104.
[12]NOWAK G.Approximation properties for generalized q-Bernstein polynomials[J].Journal of Mathematical Analysis & Applications,2009,350(1):50-55.
[13]SIMEONOV P,GOLDMAN R.Quantum Bernstein bases and quantum Bezier curves[J].Journal of Computational and Applied Mathematics,2015,288:284-303.
[14]SIMEONOV P,GOLDMAN R.Two essential properties of(q,h)-Bernsteiein Bezier curves[J].Applied Numerical Mathematics:Transactions of IMACS,2015,96:82-93.
[15]BAASS K G.The Use of Clothoid Templates in Highway Design [J].Transportation Forum,1984,1(1):47-52.
[16]YANG X X,ZHOU W W,ZHANG Y.A re-planning path correction method for collision avoidance based on PH spiral [J].Flight Dynamics,2016,34(5):86-90.
[17]SAPIDIS N S,FREY W H.Controlling the curvature of a quadratic Bézier curve [J].Computer Aided Geometric Design,1992,9(2):85-91.
[18]FREY W H,FIELD D A.Designing Bézier conic segments with monotone curature[J].Computer Aided Geometric Design,2000,17(6):457-483.
[19]CAI H H,LIU B X,CHENG Y.Transition Curve between Pa-rallel Lines Based on Bézier Curve[J].Journal of Graphics,2015,36(3):363-366.
[20]AHMAD A,GOBITHAASAN R U.Rational Quadratic Bézier Spirals [J].Sains Malaysiana,2018,47(9):2205-2211.
[21]WANG A Z,ZHAO G,HOU F.Constructing Bézier curves with monotone curvature [J].Journal of Computational and Applied Mathematics,2019,355:1-10.
[22]HE C,ZHAO G,WANG A Z,et al.Typical curve with G1 constraints for curve completion[J].Visual Computing for Industry,Biomedicine,and Art,2021,4(1):4-28.
[23]LIANG J N,XIE B,HAN L W.Combinatorial quadratic Phillips q-Bézier curves with monotone curvature[J].Journal of Gra-phics,2022,43(3):443-452.
[24]HAN L W,CHU Y,QIU Z Y.Generalized Bézier curves and surfaces based on Lupaş q-analogue of Bernstein operator[J].Journal of Computational and Applied Mathematics,2014,261:318-329.
[1] LI Hui, LI Wengen, GUAN Jihong. Dually Encoded Semi-supervised Anomaly Detection [J]. Computer Science, 2023, 50(7): 53-59.
[2] ZHU Wentao, LIU Wei, LIANG Shangsong, ZHU Huaijie, YIN Jian. Variational Continuous Bayesian Meta-learning Based Algorithm for Recommendation [J]. Computer Science, 2023, 50(7): 66-71.
[3] HUO Weile, JING Tao, REN Shuang. Review of 3D Object Detection for Autonomous Driving [J]. Computer Science, 2023, 50(7): 107-118.
[4] YAN Mingqiang, YU Pengfei, LI Haiyan, LI Hongsong. Arbitrary Image Style Transfer with Consistent Semantic Style [J]. Computer Science, 2023, 50(7): 129-136.
[5] ZHOU Bo, JIANG Peifeng, DUAN Chang, LUO Yuetong. Study on Single Background Object Detection Oriented Improved-RetinaNet Model and Its Application [J]. Computer Science, 2023, 50(7): 137-142.
[6] ZHAO Ran, YUAN Jiabin, FAN Lili. Medical Ultrasound Image Super-resolution Reconstruction Based on Video Multi-frame Fusion [J]. Computer Science, 2023, 50(7): 143-151.
[7] DAI Xuesong, LI Xiaohong, ZHANG Jingjing, QI Meibin, LIU Yimin. Unsupervised Domain Adaptive Pedestrian Re-identification Based on Counterfactual AttentionLearning [J]. Computer Science, 2023, 50(7): 160-166.
[8] HU Jiawei, JIA Zequn, SUN Yantao, LIU Qiang. Survey of Analysis and Solutions for Multi-UAV Cooperative Mission Planning Problem Under Multi-constraint Conditions [J]. Computer Science, 2023, 50(7): 176-193.
[9] JIANG Linpu, CHEN Kejia. Self-supervised Dynamic Graph Representation Learning Approach Based on Contrastive Prediction [J]. Computer Science, 2023, 50(7): 207-212.
[10] MAO Huihui, ZHAO Xiaole, DU Shengdong, TENG Fei, LI Tianrui. Short-term Subway Passenger Flow Forecasting Based on Graphical Embedding of Temporal Knowledge [J]. Computer Science, 2023, 50(7): 213-220.
[11] ZHU Yuying, GUO Yan, WAN Yizhao, TIAN Kai. New Word Detection Based on Branch Entropy-Segmentation Probability Model [J]. Computer Science, 2023, 50(7): 221-228.
[12] LI Rongchang, ZHENG Haibin, ZHAO Wenhong, CHEN Jinyin. Data Reconstruction Attack for Vertical Graph Federated Learning [J]. Computer Science, 2023, 50(7): 332-338.
[13] YU Kai, SU Tianrui. Modeling and Simulation of Point-to-Point Propagation of False Information Based on Information Risk Perception [J]. Computer Science, 2023, 50(7): 376-385.
[14] LUO Haiwen, WU Yangjun, SHANG Honghui. Many-core Optimization Method for the Calculation of Ab initio Polarizability [J]. Computer Science, 2023, 50(6): 1-9.
[15] JIN Jiexi, XIE Hehu, DU Peibing, QUAN Zhe, JIANG Hao. QR Decomposition Based on Double-double Precision Gram-Schmidt Orthogonalization Method [J]. Computer Science, 2023, 50(6): 45-51.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
渝ICP备16013121号-4
Add:18# Honghu West Rd., Yubei ,Chongqing,China    Post Code: 401121
Tel: 023-63500828/023-67039612/023-67039623
E-mail: jsjkx12@163.com
Powered by Beijing Magtech Co., Ltd.