平面几何图形相似性的代数刻画  

Algebraic characterizations of similarity of some planar geometrical graphs

作　　者：张海妮[1,2] 

机构地区：[1]陕西交通职业技术学院基础部,陕西西安710021 [2]陕西师范大学数学与信息科学学院,陕西西安710062

出　　处：《纺织高校基础科学学报》2009年第1期33-37,共5页Basic Sciences Journal of Textile Universities

基　　金：国家自然科学基金资助项目(10571113)

摘　　要：证明了映射T:R2→R2是自相似映射,即TX=k·XU+T0(X∈R2),其中U为实的2阶正交矩阵,k>0为实数,当且仅当存在常数k>0使得d(TX,TY)=k·d(X,Y)(X,Y∈R2);证明了平面上所有自相似映射之集关于复合运算成为一个群,且平面上的所有自相似压缩映射之集是一个半群;平面上的任一自相似映射T将平面上的任一条直线映射为直线且平面上任一区域D在T下的像T(D)也是一个平面区域.作为应用,研究了平面图形的相似性.It is proved that a map T： R^2→R^2 is a self-similar mapping, that is, of the form TX=k · XU ＋ TO （ VX∈R2）,where U is a 2 by 2 real orthogonal matrix and k 〉 0 is a real number, if and only if d（TX,TY） = k · d（X,Y） （ X,Y ∈ R^2） .It is also shown that the set of all self-similar mappings on the plane is a group with respect to the composition of matrices and the set of all self-similar contractive mappings on the plane is a semigroup. Moreover, it is estab- lished that every self-similar mapping on the plane maps a straight line onto a straight line and a region onto a region. As applications, algebraic characterizations of similarity of some planar geometrical graphs are obtained.

关 键 词：自相似映射 代数刻画 相似性 平面几何图形 

分 类 号：O177.1[理学—数学]