正交变换另析

THE ORTHOGONAL TRANSFORMATION

本文主要研究二维空间和三维空间的正交变换。在二维空间中,从平面上关于直线的轴对称变换谈起,把平面上的平移变换看作平面上对称轴平行的轴对称变换之偶数乘积,旋转变换看作对称轴相交于一点的轴对称变换之积,中心对称变换看作是两对称轴垂直的轴对称变换之积,并用代数法加以证明。在三维空间,亦可把关于直线的轴对称变换看作过此直线的两垂直平面为对称面的关于平面的对称之积,也用代数方法加以证明。

In this paper, we are mainly discussing the orthogonal transformation in the 2—dimensional space and 3—dimensional space, starting with the axial symmetric transformation of straight line on the plane inthe 2—dimensional space, we can consider parallel displacoment transformation as the product, which must be even times, of point symmetric transformations whose symmetric axes are parallel on the plane, rotation transformation as the product of axial symmetric transformation whose symmetric axes cross one point, central symmetric transformation as the product, which must be two times, of axial symmetric transformation, whose two symmetric axes are orthogonal. At the same time, we give all these proof with analytic method. In the 3—dimensional space, we can also consider the symmetry of straight line as the product, which must even times, of two orthogonal planes crossed through above the straight line with plane symmetry, we also give the proof with analylic method.