Quaternions for Computer Graphics by John Vince

Author:John Vince
Language: eng
Format: epub, pdf
ISBN: 9781447175094
Publisher: Springer London

Observe from Hamilton’s rules how the occurrence of ij is replaced by k. The extra imaginary k term is key to the cyclic patterns , , and , which are very similar to the cross product of two unit Cartesian vectors:

In fact, this similarity is no coincidence, as Hamilton also invented the scalar and vector products. However, although quaternions provided an algebraic framework to describe vectors, one must acknowledge that vectorial quantities had been studied for many years prior to Hamilton.

Hamilton also saw that the terms could represent three Cartesian unit vectors , and , which had to possess imaginary qualities. i.e. , etc., which didn’t go down well with some mathematicians and scientists who were suspicious of the need to involve so many imaginary terms.

Hamilton’s motivation to search for a 3-D equivalent of complex numbers was part algebraic, and part geometric. For if a complex number is represented by a couple and is capable of rotating points on the plane by , then perhaps a triple rotates points in space by . In the end, a triple had to be replaced by a a quadruple—a quaternion.

One can regard Hamilton’s rules from two perspectives. The first, is that they are an algebraic consequence of combining three imaginary terms. The second, is that they reflect an underlying geometric structure of space. The latter interpretation was adopted by P. G. Tait, and outlined in his book An Elementary Treatise on Quaternions. Tait’s approach assumes three unit vectors aligned with the x-, y-, z-axes respectively:The result of the multiplication of into or is defined to be the turning of through a right angle in the plane perpendicular to in the positive direction, in other words, the operation of on turns it round so as to make it coincide with ; and therefore briefly .

To be consistent it is requisite to admit that if instead of operating on had operated on any other unit vector perpendicular to in the plane yz, it would have turned it through a right-angle in the same direction, so that can be nothing else than .

Extending to other unit vectors the definition which we have illustrated by referring to , it is evident that operating on must bring it round to , or . [4]

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