一类保细节特征的双参数m重融合型细分
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摘要
基于B样条的光滑性,利用Laurent多项式与细分生成多项式之间的关系,构造出可生成一类m重融合型细分格式的Laurent多项式.构造的Laurent多项式不仅包含了较多经典格式,还可衍生出C~3连续且保持细节特征的新格式;特别地,分析了双参数四点三重融合型格式的支集和连续性,并给出和证明了格式C~3连续的充分必要条件.最后通过大量数值实例展示了参数对极限曲线的影响;对比图例表明,文中格式生成的极限曲线能较好地保持细节特征.
Based on the smoothness of B-splines, a Laurent polynomial which can generate a class of m-ary combined subdivision schemes is constructed by the relationship between Laurent polynomials and generated polynomials. The new Laurent polynomial can not only contain some classical subdivision schemes but also generalize C3-continuous new-type schemes with detailed features. Specifically, the support and continuities of the combined four-point ternary scheme has been analyzed; the necessary and sufficient condition for its C3 continuity are also given and proved. Plenty of numerical examples are given to illustrate the influence of parameters on the limit curves. The comparisons show that the limit curves generated by the scheme can keep the detail features well.
Based on the smoothness of B-splines, a Laurent polynomial which can generate a class of m-ary combined subdivision schemes is constructed by the relationship between Laurent polynomials and generated polynomials. The new Laurent polynomial can not only contain some classical subdivision schemes but also generalize C3-continuous new-type schemes with detailed features. Specifically, the support and continuities of the combined four-point ternary scheme has been analyzed; the necessary and sufficient condition for its C3 continuity are also given and proved. Plenty of numerical examples are given to illustrate the influence of parameters on the limit curves. The comparisons show that the limit curves generated by the scheme can keep the detail features well.
引文
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[2]Dyn N,Levin D,Gregory J A.A 4-point interpolatory subdivision scheme for curve design[J].Computer Aided Geometric Design,1987,4(4):257-268
[3]Hassan M F,Dodgson N A.Ternary and three-point univariate subdivision schemes[J].Journal of Parallel and Distributed Computing,2001,74(3):2166-2179
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[5]Huang Zhangjin.Continuity analysis and construction of uniform stationary univariate subdivision schemes[J].Journal of Software,2006,17(3):559-567(in Chinese)(黄章进.单变量均匀静态细分格式的连续性分析和构造[J].软件学报,2006,17(3):559-567)
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[7]Wang Dong,Zhang Xi,Li Guiqing,Blending subdivision curves and their applications[J].Journal of Computer-Aided Design&Computer Graphics,2007,19(3):286-291(in Chinese)(王栋,张曦,李桂清.混合细分曲线及其应用[J].计算机辅助设计与图形学学报,2007,19(3):286-291)
[8]Lin S J,You F,Luo X N,et al.Deducing interpolating subdivision schemes from approximating subdivision schemes[J].ACM Transactions on Graphics,2008,27(5):Article No.146
[9]Deng Chongyang,Wang Guozhao.A convexity preserving subdivision scheme for curve interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2009,21(8):1042-1046(in Chinese)(邓重阳,汪国昭.曲线插值的一种保凸细分方法[J].计算机辅助设计与图形学学报,2009,21(8):1042-1046)
[10]Qi Wanfeng,Luo Zhongxuan,Fan Xin.Subdivision schemes induced by approximating schemes[J].Scientia Sinica:Mathematica,2014,44(7):755-768(in Chinese)(亓万锋,罗钟铉,樊鑫.基于逼近型细分的诱导细分格式[J].中国科学:数学,2014,44(7):755-768)
[11]Luo Xiaonan,Lin Shujin,Chen Qiaozhen.A surface modeling method combining interpolation and approximating subdivision:China,101408991[P].2009-04-15(in Chinese)(罗笑南,林淑金,陈巧珍.一种插值型细分和逼近型细分相融合的曲面造型方法:中国,101408991[P].2009-04-15)
[12]Pan J,Lin S J,Luo X N.A combined approximating and interpolating subdivision scheme with C2 continuity[J].Applied Mathematics Letters,2012,25(12):2140-2146
[13]Rehan K,Sabri M A.A combined ternary 4-point subdivision scheme[J].Applied Mathematics and Computation,2016,276:278-283
[14]Tan Jieqing,Wang Bo,Xia Chenglin,et al.A class of binary subdivision schemes derived by the Laurent polynomial with parameters[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(12):2082-2087(in Chinese)(檀结庆,汪钵,夏成林,等.一类由Laurent多项式诱导的带参数二重细分[J].计算机辅助设计与图形学学报,2016,28(12):2082-2087)
[15]Siddiqi S S,Rehan K.Improved binary four point subdivision scheme and new corner cutting scheme[J].Computer and Mathematics with Applications,2010,59(8):2647-2657
[16]Charina M,Conti C.Polynomial reproduction of multivariate scalar subdivision schemes[J].Journal of Computational and Applied Mathematics,2013,240:51-61
[17]Cavaretta A S,Dahman W,Micchelli C A.Stationary subdivision[M].Providence:Memoirs of the American Mathematical Society,1991,93(453):1-186
[18]Lian J A.On a-ary subdivision for curve design:II.3-point and5-point interpolatory schemes[J].Applications and Applied Mathematics:An International Journal,2009,3(2):18-29
[19]Mustafa G,Ghaffar A,Khan F.The odd-point ternary approximating schemes[J].American Journal of Computational Mathematics,2011,1:111-118
[20]Tan Jieqing,Tong Guangyue,Zhang Li.An approximating subdivision based on interpolating subdivision scheme[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1162-1166(in Chinese)(檀结庆,童广悦,张莉.基于插值细分的逼近细分法[J].计算机辅助设计与图形学学报,2015,27(7):1162-1166)
[21]Gori L,Pitolli F,Santi E.On a class of shape-preserving refinable functions with dilation 3[J].Journal of Computational and Applied Mathematics,2013,245:62-74
[22]Tan J Q,Zhuang X L,Zhang L.A new four-point shape-preserving C3 subdivision scheme[J].Computer Aided Geometric Design,2014,31(1):57-62
[23]Novara P,Romani L.Complete characterization of the regions of C2 and C3 convergence of combined ternary 4-point subdivision schemes[J].Applied Mathematics Letters,2016,62:84-91
[24]Cao H J,Tan J Q.A binary five-point relaxation subdivision scheme[J].Journal of Information and Computational Science,2013,10(18):5903-5910
[25]Zheng Hongchan,Ye Zhenglin,Zhao Hongxing.A class of four-point subdivision scheme with two parameters and its properties[J].Journal of Computer-Aided Design&Computer Graphics,2004,16(8):1140-1145(in Chinese)(郑红婵,叶正麟,赵红星.双参数四点细分法及其性质[J].计算机辅助设计与图形学学报,2004,16(8):1140-1145)
[2]Dyn N,Levin D,Gregory J A.A 4-point interpolatory subdivision scheme for curve design[J].Computer Aided Geometric Design,1987,4(4):257-268
[3]Hassan M F,Dodgson N A.Ternary and three-point univariate subdivision schemes[J].Journal of Parallel and Distributed Computing,2001,74(3):2166-2179
[4]Hassan M F,Ivrissimitzis I P,Dodgson N A,et al.An interpolating 4-point C2 ternary stationary subdivision scheme[J].Computer Aided Geometric Design,2002,19(1):1-18
[5]Huang Zhangjin.Continuity analysis and construction of uniform stationary univariate subdivision schemes[J].Journal of Software,2006,17(3):559-567(in Chinese)(黄章进.单变量均匀静态细分格式的连续性分析和构造[J].软件学报,2006,17(3):559-567)
[6]Shen Liyong,Huang Zhangjin.A class of curve subdivision scheme with several parameters[J].Journal of Computer-Aided Design&Computer Graphics,2007,19(4):468-472(in Chinese)(申立勇,黄章进.一类多参数的曲线细分格式[J].计算机辅助设计与图形学学报,2007,19(4):468-472)
[7]Wang Dong,Zhang Xi,Li Guiqing,Blending subdivision curves and their applications[J].Journal of Computer-Aided Design&Computer Graphics,2007,19(3):286-291(in Chinese)(王栋,张曦,李桂清.混合细分曲线及其应用[J].计算机辅助设计与图形学学报,2007,19(3):286-291)
[8]Lin S J,You F,Luo X N,et al.Deducing interpolating subdivision schemes from approximating subdivision schemes[J].ACM Transactions on Graphics,2008,27(5):Article No.146
[9]Deng Chongyang,Wang Guozhao.A convexity preserving subdivision scheme for curve interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2009,21(8):1042-1046(in Chinese)(邓重阳,汪国昭.曲线插值的一种保凸细分方法[J].计算机辅助设计与图形学学报,2009,21(8):1042-1046)
[10]Qi Wanfeng,Luo Zhongxuan,Fan Xin.Subdivision schemes induced by approximating schemes[J].Scientia Sinica:Mathematica,2014,44(7):755-768(in Chinese)(亓万锋,罗钟铉,樊鑫.基于逼近型细分的诱导细分格式[J].中国科学:数学,2014,44(7):755-768)
[11]Luo Xiaonan,Lin Shujin,Chen Qiaozhen.A surface modeling method combining interpolation and approximating subdivision:China,101408991[P].2009-04-15(in Chinese)(罗笑南,林淑金,陈巧珍.一种插值型细分和逼近型细分相融合的曲面造型方法:中国,101408991[P].2009-04-15)
[12]Pan J,Lin S J,Luo X N.A combined approximating and interpolating subdivision scheme with C2 continuity[J].Applied Mathematics Letters,2012,25(12):2140-2146
[13]Rehan K,Sabri M A.A combined ternary 4-point subdivision scheme[J].Applied Mathematics and Computation,2016,276:278-283
[14]Tan Jieqing,Wang Bo,Xia Chenglin,et al.A class of binary subdivision schemes derived by the Laurent polynomial with parameters[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(12):2082-2087(in Chinese)(檀结庆,汪钵,夏成林,等.一类由Laurent多项式诱导的带参数二重细分[J].计算机辅助设计与图形学学报,2016,28(12):2082-2087)
[15]Siddiqi S S,Rehan K.Improved binary four point subdivision scheme and new corner cutting scheme[J].Computer and Mathematics with Applications,2010,59(8):2647-2657
[16]Charina M,Conti C.Polynomial reproduction of multivariate scalar subdivision schemes[J].Journal of Computational and Applied Mathematics,2013,240:51-61
[17]Cavaretta A S,Dahman W,Micchelli C A.Stationary subdivision[M].Providence:Memoirs of the American Mathematical Society,1991,93(453):1-186
[18]Lian J A.On a-ary subdivision for curve design:II.3-point and5-point interpolatory schemes[J].Applications and Applied Mathematics:An International Journal,2009,3(2):18-29
[19]Mustafa G,Ghaffar A,Khan F.The odd-point ternary approximating schemes[J].American Journal of Computational Mathematics,2011,1:111-118
[20]Tan Jieqing,Tong Guangyue,Zhang Li.An approximating subdivision based on interpolating subdivision scheme[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1162-1166(in Chinese)(檀结庆,童广悦,张莉.基于插值细分的逼近细分法[J].计算机辅助设计与图形学学报,2015,27(7):1162-1166)
[21]Gori L,Pitolli F,Santi E.On a class of shape-preserving refinable functions with dilation 3[J].Journal of Computational and Applied Mathematics,2013,245:62-74
[22]Tan J Q,Zhuang X L,Zhang L.A new four-point shape-preserving C3 subdivision scheme[J].Computer Aided Geometric Design,2014,31(1):57-62
[23]Novara P,Romani L.Complete characterization of the regions of C2 and C3 convergence of combined ternary 4-point subdivision schemes[J].Applied Mathematics Letters,2016,62:84-91
[24]Cao H J,Tan J Q.A binary five-point relaxation subdivision scheme[J].Journal of Information and Computational Science,2013,10(18):5903-5910
[25]Zheng Hongchan,Ye Zhenglin,Zhao Hongxing.A class of four-point subdivision scheme with two parameters and its properties[J].Journal of Computer-Aided Design&Computer Graphics,2004,16(8):1140-1145(in Chinese)(郑红婵,叶正麟,赵红星.双参数四点细分法及其性质[J].计算机辅助设计与图形学学报,2004,16(8):1140-1145)