利用四元数的运算法则推导球面三角公式

基于四元数的运算法则,引入向量的共轭和逆概念及2个向量之间的另一种乘法运算——格拉斯曼乘积。结果表明,球面上大圆弧对应的四元数为球心指向大圆弧终点的单位向量与球心指向大圆弧始点的单位向量逆的格拉斯曼乘积;球面角对应的四元数为终大圆弧(指向顶点)平面法向的单位向量与始大圆弧(背向顶点)平面法向的单位向量逆的格拉斯曼乘积。利用四元数运算法则可方便地推导球面三角形边或角正弦和余弦公式及第一和第二五元素公式。

Derivation of Spherical Triangle Formula Using Quaternion Algorithm

Based on the quaternion algorithm, we introduce the concepts of the conjugate and inverse of vectors and the Glassman product, which represents another multiplication of two vectors. The quaternion corresponding to a large arc on a sphere is the inverse of Glassman product of the unit vector, whose center points to the end of the large arc, and the unit vector whose center points to the beginning of the large arc. The quaternion corresponding to spherical angle is the inverse of Glassman product of the unit vector of normal plane of the final arc(toward the vertex) and the unit vector of plane normal of the initial arc (away from the vertex). It is easy to derive spherical trigonometry formulae using the quaternion algorithm, including sinusoidal and cosine formulae, and the first and second five elements formulae.