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CLC number: TP391.7

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Received: 2008-04-08

Revision Accepted: 2008-06-23

Crosschecked: 2009-02-09

Cited: 9

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Journal of Zhejiang University SCIENCE A 2009 Vol.10 No.4 P.562-569

http://doi.org/10.1631/jzus.A0820267


A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter


Author(s):  Hua-hui CAI, Guo-jin WANG

Affiliation(s):  Institute of Computer Graphics and Image Processing; more

Corresponding email(s):   chh@zju.edu.cn, gjwang@hzcnc.com

Key Words:  Transition curve, C-Bé, zier curve, Monotone curvature, Shape parameter


Hua-hui CAI, Guo-jin WANG. A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter[J]. Journal of Zhejiang University Science A, 2009, 10(4): 562-569.

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DOI - 10.1631/jzus.A0820267


Abstract: 
We constructed a single c-Bé;zier curve with a shape parameter for G2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bézier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples. This method has the following three advantages: (1) the pattern is unified; (2) the parameter able to adjust the shape of the transition curve is available; (3) the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[3] Guggenheimer, H., 1963. Differential Geometry. McGraw-Hill, New York, p.48-52.

[4] Habib, Z., 2004. Spiral Function and Its Application in CAGD. PhD Thesis, Kagoshima University, Japan.

[5] Habib, Z., Sakai, M., 2007. G2 Pythagorean hodograph quintic transition between two circles with shape control. Computer Aided Geometric Design, 24(5):252-266.

[6] Hartman, P., 1957. The highway spiral for combining curves of different radii. Trans. Am. Soc. Civ. Eng., 122:389-409.

[7] Meek, D.S., Walton, D.J., 1989. Use of Cornu spirals in drawing planar curves of controlled curvature. J. Comput. Appl. Math., 25(1):69-78.

[8] Polya, G., Szego, G., 2004. Problems and Theorems in Analysis II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry. Springer, New York, p.36-51.

[9] Walton, D.J., Meek, D.S., 1996a. A planar cubic Bézier spiral. J. Comput. Appl. Math., 72(1):85-100.

[10] Walton, D.J., Meek, D.S., 1996b. A Pythagorean hodograph quintic spiral. Computer-Aided Design, 28(12):943-950.

[11] Walton, D.J., Meek, D.S., 1999. Planar G2 transition between two circles with a fair cubic Bézier curve. Computer-Aided Design, 31(14):857-866.

[12] Walton, D.J., Meek, D.S., 2002. Planar G2 transition with a fair Pythagorean hodograph quintic curve. J. Comput. Appl. Math., 138(1):109-126.

[13] Zhang, J.W., 1996. C-curves: an extension of cubic curves. Computer Aided Geometric Design, 13(3):199-217.

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