Abstract
This paper describes a method for joining two circles with a C-shaped and an S-shaped transition curve, composed of a cubic Bézier segment. As an extension of our previous work; we show that a single cubic curve can be used for blending or for a transition curve preserving G 2 continuity regardless of the distance of their centers and magnitudes of the radii which is an advantage. Our method with shape parameter provides freedom to modify the shape in a stable manner.
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Farin, G.: Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, 4th edn. Academic, New York (1997)
Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM J. Res. Develop. 34(5), 736–752 (1990)
Habib, Z., Sakai, M.: Spiral transition curves and their applications. Sci. Math. Jpn. 61(2), 195–206. e2004, 251–262. http://www.jams.or.jp/scmjol/2004.html (2005)
Habib, Z., Sakai, M.: G 2 Pythogorean hodograph quintic transition between circles with shape control. Comput. Aided Geom. Des. 24(5), 252–266. doi:10.1016/j.cagd.2007.03.004 (2007)
Habib, Z., Sakai, M.: On PH quintic spirals joining two circles with one circle inside the other. Comput.-Aided Des. 39(2), 125–132. doi:10.1016/j.cad.2006.10.006 (2007)
Habib, Z., Sakai, M.: Transition between concentric or tangent circles with a single segment of G 2 PH quintic curve. Comput. Aided Geom. Des. 25(4-5), 247–257. doi:10.1016/j.cagd.2007.10.006 (2008)
Habib, Z., Sarfraz, M., Sakai, M.: Rational cubic spline interpolation with shape control. Comput. Graph. 29(4), 594–605. doi:10.1016/j.cag.2005.05.010 (2005)
Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1988)
Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design (Translation by L.L. Schumaker). A. K. Peters, Wellesley (1993)
Li, Z., Ma, L., Zhao, M., Mao, Z.: Improvement construction for planar g 2 transition curve between two separated circles. In: Proceedings of 6th International Conference on Computational Science—ICCS, UK, Part II. LNCS, vol. 3992, pp. 358–361. Springer, Berlin. doi:10.1007/11758525 (2006)
Li, Z., Meek, D., Walton, D.: A smooth, obstacle-avoiding curve. Comput. Graph. 30(4), 581–587 (2006)
Liang, T., Liu, J., Hung, G., Chang, Y.: Practical and flexible path planning for car-like mobile robot using maximal-curvature cubic spiral. Robot. Auton. Syst. 52(4), 312–335 (2005)
Sakai, M.: Inflection points and singularities on planar rational cubic curve segments. Comput. Aided Geom. Des. 16(3), 149–156 (1999)
Sakai, M.: Osculatory interpolation. Comput. Aided Geom. Des. 18(8), 739–750 (2001)
Sarfraz, M.: Curve fitting for large data using rational cubic splines. Int. J. Comput. Their Appl. 10(3) (2003)
Sarfraz, M. (ed.): Geometric Modeling: Techniques, Applications, Systems and Tools. Kluwer, Deventer (2004)
Sarfraz, M., Khan, M.: Automatic outline capture of arabic fonts. Int. J. Inf. Sci. 140(3–4), 269–281 (2002)
Steven, M.L.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Walton, D.J., Meek, D.S.: A planar cubic Bézier spiral. Comput. Appl. Math. 72(1), 85–100 (1996)
Walton, D.J., Meek, D.S.: A Pythagorean hodograph quintic spiral. Comput. Aided Des. 28(12), 943–950 (1996)
Walton, D.J., Meek, D.S.: Planar G 2 transition between two circles with a fair cubic Bézier curve. Comput. Aided Des. 31(14), 857–866 (1999)
Walton, D.J., Meek, D.S.: Curvature extrema of planar parametric polynomial cubic curves. Comput. Appl. Math. 134(1–2), 69–83 (2001)
Walton, D.J., Meek, D.S., Ali, J.M.: Planar G 2 transition curves composed of cubic Bézier spiral segments. Comput. Appl. Math. 157(2), 453–476 (2003)
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Dimulyo, S., Habib, Z. & Sakai, M. Fair cubic transition between two circles with one circle inside or tangent to the other. Numer Algor 51, 461–476 (2009). https://doi.org/10.1007/s11075-008-9252-1
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DOI: https://doi.org/10.1007/s11075-008-9252-1