Sains Malaysiana 47(9)(2018): 2205–2211

http://dx.doi.org/10.17576/jsm-2018-4709-31

 

Rational Quadratic Bézier Spirals

(Pilinan Kuadratik Nisbah Bézier)

 

AZHAR AHMAD^1* & R.U. GOBITHAASAN²

 

¹Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tanjung Malim, Perak Darul Ridzuan, Malaysia

 

²School of Informatics & Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu Darul Iman, Malaysia

 

Received: 10 January 2018/Accepted: 19 May 2018

 

 

ABSTRACT

A quadratic Bézier representation withholds a curve segment with free from loops, cusps and inflection points. Furthermore, this rational form provides extra freedom to generate visually pleasing curves due to the existence of weights. In this paper, we propose sufficient conditions for rational quadratic Bézier curves to possess monotonic increasing/decreasing curvatures by means of monotone curvature tests which are based on the derivative of curvature functions. We have derived a simple interval of the middle weight that assures the construction of a family of rational quadratic Bézier curves to be planar spirals, which is characterized by the turning angle, end curvatures and the chords of control polygon. The proposed formulation can be used by CAD systems for aesthetic product design, highway/railway design and robot trajectory design avoiding unwanted curvature oscillations.

 

Keywords: Bézier curves; curvature; monotonicity; rational quadratics; spirals

 

ABSTRAK

Suatu perwakilan kuadratik Bézier akan memastikan satu segmen lengkung yang bebas daripada gelung, punding serta titik lengkungbalas. Selain itu, bentuk nisbahnya pula memberikan lebih kebebasan bagi menjana lengkungan yang menyenangkan dengan sebab kewujudan pemberat. Dalam makalah ini, kami mencadangkan semua syarat yang cukup untuk suatu lengkung nisbah kuadratik Bézier untuk memiliki kelengkungan yang monoton secara meningkat/menurun melalui ujian kelengkungan monoton, yang berasaskan pembezaan fungsi kelengkungan. Kami menafsirkan satu selang mudah merujuk kepada pemberat tengah yang menjamin pembinaan satu keluarga lengkung nisbah kuadratik Bézier sebagai suatu pilinan satah. Ia dicirikan oleh sudut putaran, kelengkungan hujung dan sisi-sisi poligon kawalan. Rumus yang dicadangkan ini boleh diguna pakai pada sistem CAD untuk reka bentuk produk estetik, reka bentuk lebuhraya/landasan keretapi dan reka bentuk trajektori robot yang dapat menghindar ayunan yang tidak diingini pada kelengkungan.

 

Kata kunci: Kelengkungan; kemonotanan; kuadratik nisbah; lengkung Bezier; pilinan

REFERENCES

Ahn, Y.J. & Kim, H.O. 1998. Curvatures of the quadratic rational Bézier curves. Computers & Mathematics with Applications 36(9): 71-83.

Farin, G. 2014. Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. 5th ed. London: Academic Press.

Farin, G. 1989. Curvature continuity and offsets for piecewise conics. ACM Transactions on Graphics (TOG) 8(2): 89-99.

Farin, G., Rein, G., Sapidis, N. & Worsey, A.J. 1987. Fairing cubic B-spline curves. Computer Aided Geometric Design 4(1-2): 91-103.

Frey, W.H. & Field, D.A. 2000. Designing Bézier conic segments with monotone curvature. Computer Aided Geometric Design 17(6): 457-483.

Higashi, M., Kaneko, K. & Hosaka, M. 1988. Generation of high-quality curve and surface with smoothly varying curvature. http://dx.doi.org/10.2312/egtp.19881007. pp. 79-92.

Hu, Q. 2014. G1 approximation of conic sections by quartic Bézier curves. Computers & Mathematics with Applications 68(12): 1882-1891.

Kulfan, B. & Bussoletti, J. 2006. ‘Fundamental’ parameteric geometry representations for aircraft component shapes. In 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. p. 6948.

Lee, E.T. 1987. The rational Bézier representation for conics. In Geometric Modeling: Algorithms and New Ttrends. SIAM: Academic Press. pp. 3-19.

Li, Z., Ma, L., Meek, D., Tan, W., Mao, Z. & Zhao, M. 2006. Curvature monotony condition for rational quadratic B-spline curves. In International Conference on Computational Science and Its Applications. Springer- Berlin Heidelberg. pp. 1118-1126.

Lyche, T. & Morken, K. 2008. Spline Methods Draft. Department of Informatics, Center of Mathematics for Applications, University of Oslo, Oslo.

Pavlidis, T. 1983. Curve fitting with conic splines. ACM Transactions on Graphics (TOG) 2(1): 1-31.

Rusdi, N.A. & Yahya, Z.R. 2015. Pembinaan semula fon arab menggunakan lengkung bézier kuartik. Sains Malaysiana 44(8): 1209-1216.

Sapidis, N.S. & Frey, W.H. 1992. Controlling the curvature of a quadratic Bézier curve. Computer Aided Geometric Design 9(2): 85-91.

Sederberg, T.W. 2012. Computer aided geometric design. Computer Aided Geometric Design Course Notes. January 10.

Suenaga, K. & Sakai, M. 1999. On curvatures of rational quadratic Bézier segments. Rep. Fac. Sci. 32: 15-20.

Yoshida, N. & Saito, T. 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4(1-4): 477-486.

 

 

*Corresponding author; email: azhar.ahmad@fsmt.upsi.edu.my