齐次坐标的性质及应用.doc

摘要：在高等几何中增加了理想元素无穷远点和无穷远线，构成了理想平面，为了建立完备的一一对应关系，引入了齐次坐标。使用齐次坐标，可以简化曲线方程等的表示，为某些实际问题的计算和证明过程提供了简便的方法。解析法和齐次坐标都是解决几何问题的有力工具，但齐次坐标在一些证明计算过程中更为简便，体现了高等数学居高临下指导初等数学的原则，开阔了我们的解题思路。

关键词：齐次坐标；德萨格定理；配极原则；二次曲线

Abstract: In higher geometry, two factors are added, namely the ideal element infinite point and the line at infinity, to constitute the ideal plane.Meanwhile, in order to establish a sound one-to-one relationship, homogeneous coordinates is introduced. The use of homogeneous coordinates simplifies the curve equation, providing a concise way for calculation and proving process of some practical problems. Both the analytical method and homogeneous coordinates are powerful tools to solve geometric problems, but in some process the latter is easier, which embodies the principle of higher mathematics condescendingly guiding elementary mathematics, and broadens our problem-solving ideas.

Key words: Homogeneous coordinate; Desargues theorem; Polarity principle; Quadratic curve

目录

1.引言

2.齐次坐标的性质与应用

2.1 齐次坐标的性质

2.2齐次坐标的应用

2.2.1 德萨格定理的证明

2.2.2 配极原则的证明

2.2.3 二次曲线的相关计算

2.2.3.1二次曲线的中心

2.2.3.2直径与共轭直径

2.2.3.3渐近线

2.3齐次坐标在图形学中的应用

3.结论

参考文献

致谢