从四维转动的一般公式推导三维转动的欧拉-罗德里格斯公式
Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations
摘要：
通过使一般4D旋转矩阵的最左顶部元素（a00）等于1，一般4D旋转矩阵被专用于一般3D旋转矩阵。它的左、右四元数分量乘积的关联矩阵也相应地被特殊化。利用三维空间中坐标轴位移角度不等式证明了左右四元数互为逆元，从而证明了欧拉-罗德里格斯公式。一个一般的程序来确定一个给定的三维旋转矩阵的欧拉参数的草图。通过在一般的4D旋转矩阵中使最左边的顶部元素等于-1而不是+1，证明了3D旋转反射的欧拉-罗德里格斯公式的对应物。关键词：欧拉-罗德里格斯公式，欧拉参数，四元数，四维旋转，三维旋转，旋转反射
英文摘要：
The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised correspondingly. Inequalities involving the angles through which the coordinate axes in 3D space are displaced are used to prove that the left-hand and the right-hand quaternions are each other's inverses, thus proving the Euler-Rodrigues formula. A general procedure to determine the Euler parameters of a given 3D rotation matrix is sketched. By equating the leftmost top element to -1 instead of +1 in the general 4D rotation matrix, one proves the counterpart of the Euler-Rodrigues formula for 3D rotoreflections. Keywords: Euler--Rodrigues formula, Euler parameters, quaternions, four--dimensional rotations, three--dimensional rotations, rotoreflections
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