Quaternions and Reflections
Quaternions and Reflections
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DOI：
10.1080/00029890.1946.11991647
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发表时间：
1946-03
影响因子：
0.5
通讯作者：
H. Coxeter
中科院分区：
文献类型：
--
作者：
H. Coxeter
1.导论.这只是一个百年以来，凯莱开始使用四元数的讨论旋转。紧随其后的是布尔、唐金、克利福德、布赫海姆、克莱因、赫尔维茨、海瑟薇、斯特林汉姆和Study。显然，这些人中没有一个想到先考虑反射的简单操作，并推导出旋转是两次反射的产物。这一过程将在§§ 3和§ § 5中描述，其结果将在后面的章节中展开。每个四元数a= a0+ a1 i +~+ a3 k确定欧氏四维空间中的一个点Pa=（ao，a1，a2，aa）和一个超平面aoX 0 + a1 x1 + a2 xs + aaX 3 = 0。在超平面中的反射被发现是变换x--+-axajNa。这很容易导致经典表达式
1. Introduction. It is just a hundred years since Cayley began to use quaternions for the discussion of rotations. He was followed by Boole, Donkin, Clifford, Buchheim, Klein, Hurwitz, Hathaway, Stringham, and Study. Apparently none of these men thought of considering first the simpler operation of reflection and deducing a rotation as the product of two reflections. This procedure will be described in §§ 3 and 5, and its consequences developed in the later sections.Every quaternion a= a0+ a1i+~+ a3k determines a point Pa=(ao, a1, a2, aa) in euclidean 4-space, and a hyperplane aoXo+ a1x1+ a2xs+ aaX3= 0. The reflection in that hyperplane is found to be the transformation x--+-axajNa. This leads easily to the classical expression