一类推广的平面Koch曲线的解析表达式
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摘要
实际问题研究中一些自然现象需要用各种分形曲线来进行模拟,为了清晰了解分形曲线,就需要寻找其分形曲线的解析表达式.黄雪梅运用迭代函数系IFS来构造分形的方法研究了von Koch曲线.本文将经典Koch曲线迭代系统中的等边三角形推广到任意夹角的等腰三角形,并利用相似压缩变换,即通过不间断地将区间[0,1]进行等分迭代的方法,得到了该推广形式的Koch曲线的解析表达式,并验证了经典Koch曲线的解析表达式是本文结果的特例.
To Study some natural phenomena,it requires that use fractal curves to simulate. For better understanding fractal curves,it needs to obtain the analytical expression of fractal curves. By means of iterated function systems( IFS),Huang Xuemei research on the von Kochcurve. This paper extend equilateral triangles to isosceles triangles which have free angles between the bottom and the waist in iterated systems of the classical Koch curves,and by virtue of similar compression transformation,that is,uninterrupted uniform iteration for the interval,we get the analytical expression of generalizations of classical Koch curves and verify that analytical expression of classical Koch curves is a special case of the above result.
To Study some natural phenomena,it requires that use fractal curves to simulate. For better understanding fractal curves,it needs to obtain the analytical expression of fractal curves. By means of iterated function systems( IFS),Huang Xuemei research on the von Kochcurve. This paper extend equilateral triangles to isosceles triangles which have free angles between the bottom and the waist in iterated systems of the classical Koch curves,and by virtue of similar compression transformation,that is,uninterrupted uniform iteration for the interval,we get the analytical expression of generalizations of classical Koch curves and verify that analytical expression of classical Koch curves is a special case of the above result.
引文
[1]曾文曲,译.肯尼斯法尔科内.分形几何——数学基础及其应用[M].沈阳:东北大学出版社,2001.
[2]文志英.分形几何的数学基础[M].上海:上海科学技术教育出版社,1999.
[3]张元康,马际华.具有两个相似压缩比的拟Koch曲线Hausdorff测度的上界估计[J].山西师范大学学报(自然科学版),2015,29(1):23-26.
[4]时金金,程值军.Koch曲线的Hausdorff测度的估计值的改进[J].南阳师范学院学报,2010,9(12):25-26.
[5]朱智伟,莫家庆.Koch曲线的分形维数及计算机模拟[J].江西大学学报,2000(2):26-34.
[6]阮正顺,罗艾花,王小林.Koch曲线所围区域内的一类解析函数边值问题[J].数学的实践与认识,2011,41(12):235-238.
[7]梁永顺,苏维宜.Koch曲线及其分数阶微积分[J].数学学报,2011,54(2):227-240.
[8]杨晓玲,黄雪梅.Koch曲线的级数展开式[J].应用数学学报,2002,25(3):527-537.
[9]陈宁,张宇婷.用广义von Koch曲线生成构造D对称分形[J].小型微型计算机系统,2015,36(4):872-876.
[10]童宁江,古辉.Koch曲线的序数理论与新算法设计[J].科学技术与工程,2011,17(11):4045-4047.
[11]张显,吴波.等分Cantor集构造的一个基本性质[J].四川理工学院学报(自然科学版),2016,29(6):94-96.
[12]Shaoxu Xia,Bin Yao,Qiping Chen,Xinhai Yu,Qilin Wu.Composites with Koch fractal activated carbon fiber felt screens for strong microwave absorption[J].Composites Part B:Engineering,2017,16:1-7.
[13]Hossein Rajabloo,Vahid Amiri Kooshki,Homayoon Oraizi.Compact microstrip fractal Koch slot antenna with ELC coupling load for triple band application[J].AEU-International Journal of Electronics and Communications,2017,73:144-149.
[14]Ferdows B.Zarrabi,Zahra Mansouri,Navid P.Gandji,Hamed Kuhestani.Triple-notch UWB monopole antenna with fractal Koch and Tshaped stub[J].AEU-International Journal of Electronics and Communications,2016,70:64-69.
[15]黄雪梅.中的von Koch曲线的解析表达[J].云南大学学报(自然科学版),1997(3):291-297.
[2]文志英.分形几何的数学基础[M].上海:上海科学技术教育出版社,1999.
[3]张元康,马际华.具有两个相似压缩比的拟Koch曲线Hausdorff测度的上界估计[J].山西师范大学学报(自然科学版),2015,29(1):23-26.
[4]时金金,程值军.Koch曲线的Hausdorff测度的估计值的改进[J].南阳师范学院学报,2010,9(12):25-26.
[5]朱智伟,莫家庆.Koch曲线的分形维数及计算机模拟[J].江西大学学报,2000(2):26-34.
[6]阮正顺,罗艾花,王小林.Koch曲线所围区域内的一类解析函数边值问题[J].数学的实践与认识,2011,41(12):235-238.
[7]梁永顺,苏维宜.Koch曲线及其分数阶微积分[J].数学学报,2011,54(2):227-240.
[8]杨晓玲,黄雪梅.Koch曲线的级数展开式[J].应用数学学报,2002,25(3):527-537.
[9]陈宁,张宇婷.用广义von Koch曲线生成构造D对称分形[J].小型微型计算机系统,2015,36(4):872-876.
[10]童宁江,古辉.Koch曲线的序数理论与新算法设计[J].科学技术与工程,2011,17(11):4045-4047.
[11]张显,吴波.等分Cantor集构造的一个基本性质[J].四川理工学院学报(自然科学版),2016,29(6):94-96.
[12]Shaoxu Xia,Bin Yao,Qiping Chen,Xinhai Yu,Qilin Wu.Composites with Koch fractal activated carbon fiber felt screens for strong microwave absorption[J].Composites Part B:Engineering,2017,16:1-7.
[13]Hossein Rajabloo,Vahid Amiri Kooshki,Homayoon Oraizi.Compact microstrip fractal Koch slot antenna with ELC coupling load for triple band application[J].AEU-International Journal of Electronics and Communications,2017,73:144-149.
[14]Ferdows B.Zarrabi,Zahra Mansouri,Navid P.Gandji,Hamed Kuhestani.Triple-notch UWB monopole antenna with fractal Koch and Tshaped stub[J].AEU-International Journal of Electronics and Communications,2016,70:64-69.
[15]黄雪梅.中的von Koch曲线的解析表达[J].云南大学学报(自然科学版),1997(3):291-297.