矩阵在几何变换中的应用

论文总字数：7835字

摘 要

本文通过对平面上一些特殊的几何变换的研究，旨在让我们能够全面系统地掌握线性代数与几何的基本知识；深刻领会处理几何变换的思想方法；培养和提高抽象思维能力、逻辑推理能力、计算能力.突出矩阵在高等代数中的特殊地位，一方面矩阵本身有许多理论问题可以研究；另一方面它又是研究平面几何中几何变换的重要工具.

关键词：矩阵，几何平面，几何变换，应用

Abstract： Based on the study of some special geometric transform plane, in order to allow us to systematically master the basic knowledge of linear algebra and geometry; Deeply understand the ideological approach to geometry transform; cultivate and improve abstract thinking , logical reasoning , computation ability. Special prominence of matrix in Higher Algebra .A matrix itself has many theoretical problems can be studied; on the other hand, it is also an important tool in the study of geometry in plane geometry transformation.

Keywords：plane geometry, geometric transformation, linear algebra, matrix

目 录

1 引言·····································································4

2 欧氏平面的初等变换····························································4

2.1 平移变换·····································································4

2.2 旋转变换·····································································5

2.3 反射变换·····································································7

2.4 正交变换·····································································8

3 仿射平面的几何变换····························································8

3.1 仿射变换的定义·······························································8

3.2 仿射变换的性质·······························································9

2.3 仿射坐标系与仿射变换坐标················································9

4 射影平面的几何变换···························································10

4.1 射影变换的定义··························································10

4.2 射影变换的性质····························································11

4.3 射影坐标系与射影变换坐标··············································11

5 几何变换的若干应用···························································13

结论·····································································15

参考文献·····································································16

致谢·····································································17

1 引言

在大学高等代数课程的矩阵相关知识学习中，大多数老师把矩阵的基本定义、定理及其证明作为重点，往往忽视了它在其他方面的应用，这很容易造成理论与实际相脱节的情况发生．这种做法导致的直接问题就是，学生认为高等代数课程枯燥乏味，很难对其产生学习的乐趣，从而导致对高等代数这门课程的学习停留在问题的表面，而其本质性的内涵却很少有人能感悟．针对这一问题，本文尝试将矩阵应用到几何变换之中，让我们对矩阵有更深入的理解，达到提高学习兴趣的目的．

2 欧氏平面的初等变换

2.1 平移变换

定义1 将平面上的每个点都沿着同一个方向平行移动相同的距离的点变换称为上的一个平移变换，简称平移．

取定平行于平面的一个向量，定义的变换

其中是由定义的点，称为平面上的平移，为的平移向量．设在直角坐标系中，的坐标为，点和的坐标是．

图1

因为，由图1可得，平移变换在直角坐标系下的坐标变换公式：

而换成矩阵的形式可表示为

通过观察矩阵可知，行列式，显然平移变换是可逆变换．

2.2 旋转变换

定义2 在平面上取定一点，将上每一点都绕向同一个方向旋转相同角度的点的变换，称为上以为中心旋转角的一个旋转变换，记为．

设为上任一点，定义在上的变换 ，使得．

建立以中心为坐标原点的直角坐标系（见图2），点在此坐标系下的坐标为，点在此坐标系下的坐标为．

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