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| Construction of cubic C-Bézier spiral and its application in highway design |
| CAI Hua-hui1,2, WANG Guo-jin1 |
(1.Institute of Computer Images and Graphics, State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027,
China;
2. School of Information Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333403, China) |
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Abstract A planar cubic C-Bézier spiral with monotone curvature of constant sign was constructed to meet the requirement of highway design, and then the transition curves between straight lines and/or circular arcs were derived in detail. As it is done with clothoids in engineering, a single spiral is used for straight line to circular arc, two spirals suiting C-shaped or S-shaped transition for circular arc to circular arc, two spirals for straight line to straight line, and a single spiral for circular to circular arc when the latter is contained within the former. The concrete expressions for the first four cases were given. In the fifth case, the solution cannot always be obtained. Because straight line segments and circular arcs can be represented precisely by C-Bézier curves, the issues such as highway design can be handled in the system by C-Bézier model, avoiding the difficult situation for computer-aided design system to use the clothoids defined in terms of the Fresnel integral．
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Published: 26 February 2010
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三次C-Bézier螺线构造及其在道路设计中的应用
针对道路设计的工程需要,构造曲率单调且保号的平面三次C-Bézier螺线. 利用这条螺线,详细推导在道路设计等工业应用中直线和圆弧之间的过渡曲线. 如同工程中使用回旋曲线过渡一样,直线和圆弧之间用一条螺线过渡,圆弧与圆弧之间用一对C型或S型螺线过渡,两条直线之间用一对螺线过渡,当圆包含圆弧时用一条螺线过渡. 给出在前4种情况下螺线的具体表达式,第5种情况不一定有解. 由于直线、圆弧能够用C-Bézier曲线精确表示,可以在C-Bézier模式下统一处理整条道路设计问题,避免了以往采用Fresnel积分所表示的回旋曲线不适用于计算机辅助设计系统的情况．
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