保广义凸的曲线插值方法
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摘要
为了寻求简易有效的保凸曲线插值,提出一种用分段Bézier曲线拼接的方法,可以构造一条光滑的插值曲线.对于给定的平面有序点列,根据有序点列所连成的折线的运动方向,确定曲线在每个插值点处的切向量;进而利用点列广义凸的概念,在每2个相邻点之间按设计的算法直接插入2个三次Bézier曲线的控制顶点,该4点确定一条三次Bézier曲线;从而得到通过这组点列的分段光滑Bézier插值曲线,整条曲线G1连续.每段曲线的中间2个控制顶点由4个相邻的顶点确定.该方法适用于一般有序点列的插值,并具有保凸性,曲线局部形状可调,算法简单和计算量少的特点.最后通过实例说明了文中方法的有效性及正确性.
In order to find a simple and effective convex curve interpolation, this paper proposes a method of using piecewise Bézier curve to construct a smooth interpolation curve. For a given set of point sequence, according to the direction of the polyline produced by the data points, we determine the vector of the curve at the every interpolating point. And then by using the concept of generalized convexity point sequence, we obtain the control points of the piecewise Bézier curves of degree three between every two points in view of the design algorithm. This four points determine a Bézier curves of degree three. Thus, we construct the piecewise smooth Bézier curves which pass through the data points. The middle two control points of each piece curve are determined by four adjacent points. The present method is available for an arbitrary set of point sequence, and enjoys the following advantages: Convexity-preserving; Local shape adjustable; Do not need to solve a system of equations; Simple algorithm and less computation. In the end, it shows that the method is very efficient through some examples.
In order to find a simple and effective convex curve interpolation, this paper proposes a method of using piecewise Bézier curve to construct a smooth interpolation curve. For a given set of point sequence, according to the direction of the polyline produced by the data points, we determine the vector of the curve at the every interpolating point. And then by using the concept of generalized convexity point sequence, we obtain the control points of the piecewise Bézier curves of degree three between every two points in view of the design algorithm. This four points determine a Bézier curves of degree three. Thus, we construct the piecewise smooth Bézier curves which pass through the data points. The middle two control points of each piece curve are determined by four adjacent points. The present method is available for an arbitrary set of point sequence, and enjoys the following advantages: Convexity-preserving; Local shape adjustable; Do not need to solve a system of equations; Simple algorithm and less computation. In the end, it shows that the method is very efficient through some examples.
引文
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[7]Zhang Renjiang,Kan Dunzhi.Convexity-preserving interpolation using the polygonal line of data points[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1167-1171(in Chinese)(章仁江,阚敦芝.利用点列折线的保凸插值[J].计算机辅助设计与图形学学报,2015,27(7):1167-1171)
[8]Dyn N,Levin D,Liu D.Interpolatory convexity-preserving subdivision schemes for curves and surfaces[J].ComputerAided Design,1992,24(4):211-216
[9]Deng Jiansong,Feng Yuyu.A method to construct a smooth convexity-preserving parametric-like curve through interpolatory subdivision[J].Applied Mathematics A Journal of Chinese Universities,1998,13(suppl):1-8(in Chinese)(邓建松,冯玉瑜.一种平面保凸插值构造光滑曲线的方法[J].高校应用数学学报:A辑,1998,13(增刊):1-8)
[10]Albrecht G,Romani L.Convexity preserving interpolatory subdivision with conic precision[J].Applied Mathematics and Computation,2012,219(8):4049-4066
[11]Amat S,Donat R,Trillo J C.Proving convexity preserving properties of interpolatory subdivision schemes through reconstruction operators[J].Applied Mathematics and Computation,2013,219(14):7413-7421
[12]Ding Youdong.A type of nonlinear 4-point Preserving-convexity interpolatory discrete subdivision scheme and its application[J].Journal of Fudan University:Natural Science,2000,39(1):9-14(in Chinese)(丁友东.一类非线性保凸插值离散细分格式及其性质[J].复旦学报:自然科学版,2000,39(1):9-14)
[13]Dyn N,Hormann K.Geometric conditions for tangent continuity of interpolatory planar subdivision curves[J].Computer Aided Geometric Design,2012,29(6):332-347
[2]Zhang Caiming,Wang Jiaye.C2 quartic spline surface interpolation[J].Science in China:Series E,2003,33(2):116-126(in Chinese)(张彩明,汪嘉业.C2连续的四次样条曲面插值[J].中国科学:E辑,2003,33(2):116-126)
[3]Schweikert D G.An interpolation curve using a splines in tension[J].Journal of Mathematics and Physics,1966,45(1):312-317
[4]Pan Yongjuan,Wang Guojin.Monotonicity-preserving interpolation with a kind of planar parameter curve[J].Journal of Software,2003,14(8):1439-1447(in Chinese)(潘永娟,王国瑾.一类平面参数曲线的保单调插值[J].软件学报,2003,14(8):1439-1447)
[5]Manni C,Sablonnière P.Monotone interpolation of order 3 by C2 cubic splines[J].IMA Journal of Numerical Analysis,1997,17(2):305-320
[6]Romani L.A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control[J].Computer Aided Geometric Design,2010,27(1):36-47
[7]Zhang Renjiang,Kan Dunzhi.Convexity-preserving interpolation using the polygonal line of data points[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1167-1171(in Chinese)(章仁江,阚敦芝.利用点列折线的保凸插值[J].计算机辅助设计与图形学学报,2015,27(7):1167-1171)
[8]Dyn N,Levin D,Liu D.Interpolatory convexity-preserving subdivision schemes for curves and surfaces[J].ComputerAided Design,1992,24(4):211-216
[9]Deng Jiansong,Feng Yuyu.A method to construct a smooth convexity-preserving parametric-like curve through interpolatory subdivision[J].Applied Mathematics A Journal of Chinese Universities,1998,13(suppl):1-8(in Chinese)(邓建松,冯玉瑜.一种平面保凸插值构造光滑曲线的方法[J].高校应用数学学报:A辑,1998,13(增刊):1-8)
[10]Albrecht G,Romani L.Convexity preserving interpolatory subdivision with conic precision[J].Applied Mathematics and Computation,2012,219(8):4049-4066
[11]Amat S,Donat R,Trillo J C.Proving convexity preserving properties of interpolatory subdivision schemes through reconstruction operators[J].Applied Mathematics and Computation,2013,219(14):7413-7421
[12]Ding Youdong.A type of nonlinear 4-point Preserving-convexity interpolatory discrete subdivision scheme and its application[J].Journal of Fudan University:Natural Science,2000,39(1):9-14(in Chinese)(丁友东.一类非线性保凸插值离散细分格式及其性质[J].复旦学报:自然科学版,2000,39(1):9-14)
[13]Dyn N,Hormann K.Geometric conditions for tangent continuity of interpolatory planar subdivision curves[J].Computer Aided Geometric Design,2012,29(6):332-347