Quaternion Function Reference

opd

Orthogonal plane decomposition

Syntax

[p, m] = opd(q, a, b)

Description

opd(q, a ,b) decomposes a quaternion array element-by-element q into two components in orthogonal planes defined by a and b using the formulae p = ½(q + aqb) and m = ½(q - aqb). The third parameter is optional, if omitted it is set equal to the second. If a is a pure quaternion, and b is omitted, p is parallel to a, and m is in the plane normal to a. (Parallel/perpendicular decomposition.) Other cases are determined by the values of the second and third parameters. Coxeter's paper is recommended as a reference for details of the geometry.

Examples

>> q = randq;
>> [p,m] = opd(q, randv)
 
p = 2.776e-17 - 0.1426 * I - 0.8066 * J - 0.07652 * K
 
m = -0.4503 + 0.3269 * I - 0.0668 * J + 0.09476 * K
 
>> scalar_product(p,m)

ans = -3.9899e-17  % p and m are orthogonal.

See Also

QTFM function: ohd

References

  1. Eckhard Hitzer and Stephen J. Sangwine, ‘The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations’, Chapter 2, pp. 15-39 in: Eckhard Hitzer and Stephen J. Sangwine (eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Birkhäuser/Springer Basel, 2013. xxvii+338 pp. DOI: 10.1007/978-3-0348-0603-9.
  2. Ell, T. A. and Sangwine, S. J., 'Quaternion Involutions and Anti-Involutions', Computers and Mathematics with Applications, 53(1), January 2007, 137-143. DOI: 10.1016/j.camwa.2006.10.029.
  3. H. S. M. Coxeter, 'Quaternions and reflections', American Mathematical Monthly, 53(3), 136-146, 1946.

(c) 2008-2016 Stephen J. Sangwine and Nicolas Le Bihan

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