中文摘要：

四元数在三维空间基准的转换中已得到广泛应用,但其理论依据并不是很清晰。本文在理论上研究了四元数的一些基本性质,证明了坐标旋转变换等价于四元数的正交变换。利用基本四元数的定义,证明了适用于坐标旋转的所有四元数都是由若干个基本四元数的格拉斯曼乘积得到的。同时,给出了四元数与坐标旋转矩阵之间的理论关系。

英文摘要：

The quaternion has been widely used in three dimensional space datum transformation,but its theoretical basis is not very clear.This paper studies some basic properties of the quaternion theory,and it proves that coordinate rotation transformation is equivalent to the orthogonal transformation of quaternion.Using the definition of basic quaternions,it establishes that all quaternions applied to coordinate rotational transformation are made by several basic quaternion＇s Glassman product.At the same time,the theoretical relationship between quaternions and coordinate rotation matrix are given.