Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form
By Stephen J. Sangwine, N. Le Bihan. 2008
53 citations recorded by the source corpus.
Abstract.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.
Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/8bf7f0625af52d374e2c6551aad3f67d1ff50ac8
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In Applied Mathematics
- Todd A. Ell
- N. Le Bihan
- Stephen J. Sangwine
- Liming Zhang
- Chenlei Guo
- Jianyu Ou
- João Pedro Morais
- Kit Ian Kou
In Atomic and Molecular Physics, and Optics
Citations in the corpus
- Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions (2010, background): “For 8The cosine and sine of a complex angle are simply defined in terms of Euler’s formula as cos z = 1 2 ( eIz + e−Iz ) and sin z = − 1 2 I ( eIz − e−Iz ) ; or, if we write z = x+Iy: cos z = cosx cosh y−I sinx sinh y and sin z = sinx cosh y+I cosx sinh y [4, See complexes]. example: α+ Iγ = A = W .”