Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-23T18:04:55.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Cubic Spiral Transition Matching G2 Hermite End Conditions

Published online by Cambridge University Press:  28 May 2015

Zulfiqar Habib  and
Manabu Sakai
Zulfiqar Habib*
Affiliation:
COMSATS Institute of Information Technology, Department of Computer Science, Defense Road, Off Raiwind Road, Lahore, Pakistan
Manabu Sakai*
Affiliation:
Department of Mathematics & Computer Science, Koorimoto 1-21-35, Kagoshima 890-0065, Japan
*
Corresponding author.Email address:drzhabib@ciitlahore.edu.pk
Corresponding author.Email address:msakai@sci.kagoshima-u.ac.jp
Get access

Abstract

This paper explores the possibilities of very simple analysis on derivation of spiral regions for a single segment of cubic function matching positional, tangential, and curvature end conditions. Spirals are curves of monotone curvature with constant sign and have the potential advantage that the minimum and maximum curvature exists at their end points. Therefore, spirals are free from singularities, inflection points, and local curvature extrema. These properties make the study of spiral segments an interesting problem both in practical and aesthetic applications, like highway or railway designing or the path planning of non-holonomic mobile robots. Our main contribution is to simplify the procedure of existence methods while keeping it stable and providing flexile constraints for easy applications of spiral segments.

Keywords

65D0565D0765D1065D1765D18Path planningspiralcubic BézierG2 HermiteComputer-Aided Design (CAD)computational geometry
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 525 - 536
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Burchard, H.G., Ayers, J.A., Frey, W.H., and Sapidis, N.S.. Designing fair curves and surfaces, chapter Approximation with aesthetic constraints. 1993.CrossRefGoogle Scholar
[2]Dietz, D. A. and Piper, B.. Interpolation with cubic spirals. Computer Aided Geometric Design, 21(2):165–180, 2004.CrossRefGoogle Scholar
[3]Dietz, D. A., Piper, B., and Sebe, E.. Rational cubic spirals. Computer-Aided Design, 40:3–12, 2008.CrossRefGoogle Scholar
[4]Gibreel, G. M., Easa, S. M., Hassan, Y., and El-Dimeery, I. A.. State of the art of highway geometric design consistency. ASCE Journal of Transportation Engineering, 125(4) :305–313, 1999.CrossRefGoogle Scholar
[5]Guggenheimer, H.. Differential Geometry. McGraw-Hill, New York, 1963.Google Scholar
[6]Habib, Z. and Sakai, M.. G2 Pythagorean hodograph quintic transition between two circles with shape control. Computer Aided Geometric Design, 24(5):252–266, 2007.CrossRefGoogle Scholar
[7]Habib, Z. and Sakai, M.. On PH quintic spirals joining two circles with one circle inside the other. Computer-Aided Design, 39(2):125–132, 2007.CrossRefGoogle Scholar
[8]Habib, Z. and Sakai, M.. Fair path planning with a single cubic spiral segment. pp.121–125, USA, August 2008. The Proceedings of IEEE International Conference on Computer Graphics, Imaging and Visualization, Malaysia, IEEE Computer Society Press.Google Scholar
[9]Habib, Z. and Sakai, M.. Transition between concentric or tangent circles with a single segment of G 2 PH quintic curve. Computer Aided Geometric Design, 25(4-5) :247–257, 2008.CrossRefGoogle Scholar
[10]Habib, Z. and Sakai, M.. Admissible regions for rational cubic spirals matching G 2 hermite data. Computer-Aided Design, 42(12):1117–1124, 2010.CrossRefGoogle Scholar
[11]Habib, Z. and Sakai, M.. Interpolation with PH quintic spirals. USA, 2010. The Proceedings of IEEE International Conference on Computer Graphics, Imaging and Visualization, Sydney, IEEE Computer Society Press. In press.Google Scholar
[12]Henrici., PApplied and Computational Complex Analysis, volume 1. Wiley, New York, 1988.Google Scholar
[13]Kelly, A. and Nagy., B.Reactive nonholonomic trajectory generation via parametric optimal control. The International Journal of Robotics Research, 22(7/8) :583–601, 2003.Google Scholar
[14]Murray, R. M., Laumond, J. P., and Jacobs, P. E.. A motion planner for non-holonomic mobile robots. IEEE Transactions on Robotics and Automation, 10(3):577–593, 1994.Google Scholar
[15]Marjeta, K.. Geometric hermite interpolation by cubic G 1 splines. Nonlinear Analysis: Theory, Methods & Applications, 70:2614–2626, 2009.Google Scholar
[16]Raza, A. A., Habib, Z., and Sakai, M.. Interpolation with rational cubic spirals. pp.98–103, USA, October 2008. The Proceedings of 4th IEEE International Conference on Emerging Technologies, ICET-Pakistan, IEEE Computer Society Press.Google Scholar
[17]Sakai, M.. Osculatory interpolation. Computer Aided Geometric Design, 18(8):739–750, 2001.CrossRefGoogle Scholar
[18]Sarpono, D., Habib, Z., and Sakai, M.. Fair cubic transition between two circles with one circle inside or tangent to the other. Numerical Algorithms, 51(4):461–476, 2009.Google Scholar
[19]Walton, D. J. and Meek, D. S.. G 2 curve design with a pair of Pythagorean hodograph quintic spiral segments. Computer Aided Geometric Design, 24(5):267–285, 2007.CrossRefGoogle Scholar