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本文是在射影几何学及高等代数学的初步知識的基础上来写的。首先是在欧氏几何学的基础上从几何概念引进欧氏平面上的三种具体变換——运动变換、仿射变換及射影变換,并介紹它們的一些性质。然后在高等代数学抽象羣的基础上建立变換羣的概念并証明上述三种变換集合均成变換羣。有了这些具体变換羣及它們不变性貭后,最后再抽象地介紹变換羣与相应几何学的思想并說明其历史作用及近代一些发展情况。§1.平面上的运动变換 欧氏平面上的一个一一变換如果任意两点的距离与其对应两点的距离相等,则这个一一变換叫做运动变換,簡称运动。在一个一一变換里,被变換的点和图形叫原象,而它所变換成的图形叫做它的象。例如△ABC经运动得到△A′B′C′,我們說△A′B′C′是△ABC的象,而△ABC则是△A′B′C′的原象。自此定义可推出下列簡单性貭。它們均是度量性质。
This article is based on the preliminary knowledge of projective geometry and advanced mathematics. First of all, on the basis of Euclidean geometry, three specific transformations on the Euclidean plane are introduced from the concept of geometry - motion transformation, affine transformation and projective transformation, and some of their properties are introduced. Then the concept of transform group is established on the basis of abstract groups of higher mathematics and it is proved that the above three kinds of transform sets all become transform groups. With these concrete transformation groups and their invariances, we finally abstractly introduce the idea of transformation groups and corresponding geometry and explain their historical role and some recent developments. § 1. Motion transformation in the plane A one-to-one transformation in the Euclidean plane If the distance between any two points is equal to the distance between two corresponding points, this one-to-one transformation is called motion transformation, or simply motion. In a one-to-one transformation, the point and the figure being transformed are called the original, and the figure it transforms into is called its image. For example, △ABC moves to obtain ΔA’B’C’. We say that ΔA’B’C’ is the image of △ABC, and △ABC is the original image of △A’B’C’. From this definition, the following simplicity can be introduced. They are all metrics.