过异面四点确定球面方程的策略分析

Investigation on the Problem of Determining the Equation of the Sphere through Four Points in Three-dimensional Space

二次曲面理论是大学解析几何知识模块的重难点,每届中国大学生数学竞赛都会涉及对二次曲面理论的考查.通过研究第三届(2011年)中国大学生数学竞赛预赛第一题,分析并给出三维空间中过异面四点确定球面方程的策略:利用代点求参数值的思路确定球面的方程;利用球面的几何实质确定球面的方程;利用球心的特殊几何位置属性确定球面的方程;利用球面簇确定球面的方程;利用公式确定球面的方程.

The theory of quadric surface is an important and difficult point in the course of analytic geometry in universities,and every Chinese undergraduate mathematics contest involves the examination of the theory of quadric surface.After the investigation on the first question of the third Chinese undergraduate mathematics preliminary contest(held in 2011),we investigate and provide the strategy to determine equations of spheres through four points in three-dimensional Euclidean space:To determine the equation of the concerned sphere by substituting the coordinates and calculating out the involved parameters,to determine the equation of the concerned sphere by using the geometric essence of shperes,to determine the equation of the concerned sphere by using the geometric propertis of centers of shperes,to determine the equation of the concerned sphere by exploiting the idea of families of spheres,to determine the equation of the concerned sphere by applying well-kown formulae.