在三角域上构造三次多项式插值曲面
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摘要
为满足矿山地形的拟合、水流深度的绘制等很多特殊工程数据量大、有一定的光顺要求但又不需要曲面过于凸起饱满这一需求,提出一种C1连续的三次多项式插值曲面,同时有针对性地提出一种一阶偏导数估计算法.首先将空间散乱数据点投影到平面后进行三角划分;其次针对每个三角形,在其每条边上构造一个C1连续的三次多项式曲面片,由这3个曲面片加权平均形成该三角形的曲面片;最后将所有三角形上的曲面片拼合成整体曲面.为使生成的曲面尽可能地贴近数据点所建议的形状,在曲面求解过程中将数据点分成内部点和边界点分别估计偏导数.实验结果表明,该算法计算量小、具有良好的局部性,并给出了新曲面的效果.
Many special projects, such as mining shape fitting and water depth render, need large amounts of data and have certain requirement on smoothness of interpolation surfaces while the surfaces can't be too convex. To satisfy these requirements, this paper presents a C1 continue cubic polynomial interpolation surface and a target method for estimating partial derivatives. Firstly, projected the spatial scattered data points onto a plane and then triangulated them; Secondly, interpolated three piecewise cubic polynomial patches on each triangle which were C1 continue on each edge respectively, and the weighted combination of the three patches constructed surface patch of each triangle. Lastly, all surface patches of all triangles were jointed together to form the whole surface. To make the surface be closed to advised shape as far as possible, different partial derivatives estimation methods were used according to the data point was boundary point or not. New method is simple to compute and has a good local property. The result of new surface is given in experiments.
Many special projects, such as mining shape fitting and water depth render, need large amounts of data and have certain requirement on smoothness of interpolation surfaces while the surfaces can't be too convex. To satisfy these requirements, this paper presents a C1 continue cubic polynomial interpolation surface and a target method for estimating partial derivatives. Firstly, projected the spatial scattered data points onto a plane and then triangulated them; Secondly, interpolated three piecewise cubic polynomial patches on each triangle which were C1 continue on each edge respectively, and the weighted combination of the three patches constructed surface patch of each triangle. Lastly, all surface patches of all triangles were jointed together to form the whole surface. To make the surface be closed to advised shape as far as possible, different partial derivatives estimation methods were used according to the data point was boundary point or not. New method is simple to compute and has a good local property. The result of new surface is given in experiments.
引文
[1]Nielson G M.The side-vertex method for interpolation in triangles[J].Journal of Approximation Theory,1979,25(4):318-336
[2]Hagen H.Geometric surface patches without twist constraints[J].Computer Aided Geometric Design,1986,3(3):179-184
[3]Xu Lin.Infinite interpolation on triangles[J].Journal of Software,2007,18(2):430-441(in Chinese)(徐琳.三角形域上的超限插值方法[J].软件学报,2007,18(2):430-441)
[4]Mann S.An improved parametric side-vertex triangle mesh interpolant[C]//Proceedings of the Graphics Interface’98.Toronto:Canadian Human-Computer Communications Society,1998:35-42
[5]Hussain M Z,Hussain M.C1 positive scattered data interpolation[J].Computers&Mathematics with Applications,2010,59(1):457-467
[6]Wang J Y,Zhang C M.Polynomial of degree four interpolation on triangles[J].Journal of Computational Mathematics,1991,9(2):155-162
[7]Zhang Yongchun,Da Feipeng,Song Wenzhong.Piecewise C1surfaces based on bivariate quartic box-splines for arbitrary triangular meshes[J].Journal of Software,2006,17(10):2211-2220(in Chinese)(张永春,达飞鹏,宋文忠.任意三角形网格的基于二元四次箱样条分片C1曲面[J].软件学报,2006,17(10):2211-2220)
[8]Lai M C,Meile C.Scattered data interpolation with nonnegative preservation using bivariate splines and its application[J].Computer Aided Geometric Design,2015,34:37-49
[9]Yan Lanlan,Han Xuli.Trigonometric polynomial curve and surface with many advantages[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(10):1971-1979(in Chinese)(严兰兰,韩旭里.具有多种优点的三角多项式曲线曲面[J].计算机辅助设计与图形学学报,2015,27(10):1971-1979)
[10]Zhu Y P,Han X L,Quasi-Bernstein-Bézier polynomials over triangular domain with multiple shape parameters[J].Applied Mathematics and Computation,2015,250:181-192
[11]Huang Youdu,The C2 quintic interpolation over triangles[J].Mathematica Numerica Sinica,1995(2):186-195(in Chinese)(黄有度.三角域上C2连续的分片二元五次插值多项式[J].计算数学,1995(2):186-195)
[12]Gao S S,Zhang C M,Li Z.Interpolation by piecewise quadric polynomial to scattered data points[M]//Lecture Notes in Computer Science.Heidelberg:Springer,2006,4292:106-115
[13]Gao Shanshan,Zhang Caiming.Construction of fairing surface with polynomial interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2008,20(6):759-764(in Chinese)(高珊珊,张彩明.散乱数据点多项式插值光顺曲面的构造[J].计算机辅助设计与图形学学报,2008,20(6):759-764)
[14]Nagata T.Simple local interpolation of surfaces using normal vectors[J].Computer Aided Geometric Design,2005,22:327-347
[15]Franke R.A critical comparison of some methods for interpolation of scattered data[R].Monterey:Naval Postgraduate School,1979
[2]Hagen H.Geometric surface patches without twist constraints[J].Computer Aided Geometric Design,1986,3(3):179-184
[3]Xu Lin.Infinite interpolation on triangles[J].Journal of Software,2007,18(2):430-441(in Chinese)(徐琳.三角形域上的超限插值方法[J].软件学报,2007,18(2):430-441)
[4]Mann S.An improved parametric side-vertex triangle mesh interpolant[C]//Proceedings of the Graphics Interface’98.Toronto:Canadian Human-Computer Communications Society,1998:35-42
[5]Hussain M Z,Hussain M.C1 positive scattered data interpolation[J].Computers&Mathematics with Applications,2010,59(1):457-467
[6]Wang J Y,Zhang C M.Polynomial of degree four interpolation on triangles[J].Journal of Computational Mathematics,1991,9(2):155-162
[7]Zhang Yongchun,Da Feipeng,Song Wenzhong.Piecewise C1surfaces based on bivariate quartic box-splines for arbitrary triangular meshes[J].Journal of Software,2006,17(10):2211-2220(in Chinese)(张永春,达飞鹏,宋文忠.任意三角形网格的基于二元四次箱样条分片C1曲面[J].软件学报,2006,17(10):2211-2220)
[8]Lai M C,Meile C.Scattered data interpolation with nonnegative preservation using bivariate splines and its application[J].Computer Aided Geometric Design,2015,34:37-49
[9]Yan Lanlan,Han Xuli.Trigonometric polynomial curve and surface with many advantages[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(10):1971-1979(in Chinese)(严兰兰,韩旭里.具有多种优点的三角多项式曲线曲面[J].计算机辅助设计与图形学学报,2015,27(10):1971-1979)
[10]Zhu Y P,Han X L,Quasi-Bernstein-Bézier polynomials over triangular domain with multiple shape parameters[J].Applied Mathematics and Computation,2015,250:181-192
[11]Huang Youdu,The C2 quintic interpolation over triangles[J].Mathematica Numerica Sinica,1995(2):186-195(in Chinese)(黄有度.三角域上C2连续的分片二元五次插值多项式[J].计算数学,1995(2):186-195)
[12]Gao S S,Zhang C M,Li Z.Interpolation by piecewise quadric polynomial to scattered data points[M]//Lecture Notes in Computer Science.Heidelberg:Springer,2006,4292:106-115
[13]Gao Shanshan,Zhang Caiming.Construction of fairing surface with polynomial interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2008,20(6):759-764(in Chinese)(高珊珊,张彩明.散乱数据点多项式插值光顺曲面的构造[J].计算机辅助设计与图形学学报,2008,20(6):759-764)
[14]Nagata T.Simple local interpolation of surfaces using normal vectors[J].Computer Aided Geometric Design,2005,22:327-347
[15]Franke R.A critical comparison of some methods for interpolation of scattered data[R].Monterey:Naval Postgraduate School,1979