图3。尖的点出现在图上
为了满足上述条件，需要另一个算法：De Casteljau算法。
De Casteljau算法描述如何细分一个Bézier曲线曲线为两段(图4所示[8]）。控制点
P1，P2，P3，P4通过线段连接。它们再除以相同的比例t：1-t，生成点P12，P23，P34。那么那些新生成的线也按相同的t：1-t比例细分产生的新的节点，只有一个点被留下。
图4。De Casteljau算法
为了避免尖的点出现在地图上，De Casteljau算法是必需的。一般来说，尖的点总是重合点。所以采取一些措施。在三次Bézier曲线，平滑的轮廓过程中，第四点各组将改为De Casteljau的算法。添加一个新的点的中间位置在第三和第四点中的前一组。然后添加另一个点在第一和第二组第二点中间的位置(如图5所示，P34是点P3和P4的中间位置，P45是点P4和P5的中间位置）。在这种情况下，在的第一组和第二组之间
有三个点。沿着De Casteljau算法的思想，将生成一个新的曲线，通过P34和P45。其结果是相当不错的。(见图6)。
4 结论
图5。三次Bézier曲线和De Casteljau算法
图6。最后的结果 在这个过程中绘制的等值线，该算法绘制等值线由矩形网格上进行了研究。本文加入了三次Bézier曲线和De Casteljau算法的顺滑等值线，取得了良好的结果。该文献中提到的算法是很容易理解和趋向来实现的，Bézierr曲线拟合的方法具有很强的适应性。
鸣谢单位
感谢聚丰软件研究所的工作人员，四川省成都理工大学信息技术和科学与技术的基础配套工程。 参考文献
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[2] Wei Mei-yan. Research on contour’s smoothing algorithm based on the regualr grid data[J]. Modern Computer.2007(4):30~32(in Chinese).
[3] Yu Xuetao. Xi Yantao. Zhu Jincan. Niu Kun. Yu Shuyuan. Application of parabola-spline algorithm in smoothing contour[J]. Urban Geotechnical Investigation & Surveying. 2010(5):135~137(in Chinese).
[4] G. Farin. Curves and surfaces for computer aided geometric design: a practical guide. Academic Press, 1997.
[5] F. Yamaguchi. Curves and Surfaces in Computer Aided Geometric Design. Springer Verlag, 1988.
[6] Aleksas Riskus. Approximation of a cubic Bezier curve by circular arcs and vice versa. Information Technology and Control. 2006(35).
[7] Han Li-na. Geng Guo-hua. A smoothed contours generating algorithm for the multi-segment bezier curve[J]. Computer Engineering and Science. 2010,32(11) (in Chinese).
[8] Xu Tian. Liu Ling-xia. Bezier curve algorithm describing and how to draw Bezier curve by program design. Journal of Anyang Normal University. 2006(05):49~52(in Chinese).
外文原文二：
Contour Smoothing Algorithm Based on Bezier curves and application
Shu Hongping,Wei Zhaoyu
Abstract—In the process of plotting contours by rectangular grids, many polylines appeared in the contour map. To improve it, this paper uses cubic Bezier curves to fit contours.First, scan all the known points and divide them into groups of4. Then, draw cubic Bezier curves in groups. But cuspidal points turn up. And to avoid cuspidal points, de Casteljau Algorithm has been used. Add a new point at the middle position of the third and the fourth point in the previous group. Add another point at the middle position of the first and the second point in the next group. So there are three points between adjacent cubic Bezier curves. Plot another curve following the idea of de Casteljau Algorithm during these three points. Experiments show a combination of these two methods brings pretty good result.
Keywords-polylines; cubic Bezier curve; cuspidal points; de Casteljau Algorithm; smooth
I. INTRODUCTION
Contour maps are mainly used in geographical maps,engineering analysis and calculation. However, they are also useful to show isobars in weather maps, isotherms in temperature field. The algorithms to generate contours are different from each other according to their different application.
In order to draw Isothermal, the paper uses rectangle grids. Nevertheless, in the process of generating contours automatically, by linking the isoline point with line, it generates a
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