Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form
Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form
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DOI：
10.1007/s00006-008-0128-1
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发表时间：
2008-02
影响因子：
1.5
通讯作者：
S. Sangwine;N. L. Bihan
中科院分区：
文献类型：
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作者：
S. Sangwine;N. L. Bihan
受 Cayley-Dickson 表示的启发，我们提出了一种新的四元数极坐标表示。在这个新的极坐标表示中，四元数由一对复数表示，如凯莱-迪克森形式，但这里这两个复数是复数“模”和复数“参数”。与 Cayley-Dickson 形式一样，两个复数位于同一复平面（使用相同的复数根 -1），但复数相位乘以指数函数中不同的复数根 -1。我们展示了如何从笛卡尔形式的任意四元数计算“模数”和“参数”。
We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex ‘modulus’ and a complex ‘argument’. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of −1), but the complex phase is multiplied by a different complex root of −1 in the exponential function. We show how to calculate the ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form.