Quaternion Function Reference

ohd

Orthogonal hyperplane decomposition

Syntax

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

Description

ohd(q, a ,b) decomposes a quaternion array element-by-element q into two components in orthogonal hyperplanes defined by a and b using the formulae p = ½(q + acb) and m = ½(q - acb) where c is the conjugate of q. 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] = ohd(q, randq, randq)
 
p =  0.1606 - 0.1504 * I + 0.6234 * J - 0.6342 * K
 
m =  0.1713 + 0.3609 * I + 0.00905 * J - 0.0333 * K
 
>> scalar_product(p,m)

ans = -1.2490e-16  % p and m are orthogonal.

See Also

QTFM function: opd

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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