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A Quaternion Algebra Tool Set

Parts

Simple algebra

Multiplication

Trig

Powers

Logs

Multiplying by exponentials

Here is a compilation of basic algebra for quaternions. It should look very similar to complex algebra, since it contains three sets of complex numbers, t + x i, t + y j, and t + z k. To strengthen the link, and keep things looking simpler, all quaternions have been written as a pair of a scalar t and a 3-vector V, as in (t, V). All these relations have been tested in a C library and a Java quaternion calculator.

Technical note: it is vital that every tool in this set can be expressed as working with a whole quaternion q. This will make doing quaternion analysis with automorphic functions fruitful.

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Parts

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Simple algebra

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Multiplication

The Grassman product as defined here uses the same rule Hamilton developed. The Euclidean product takes the conjugate of the first of the two elements (following a tradition from quantum mechanics).

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Trigonometry

Note: since the unit vectors of sine and cosine are the same, these two commute so the order is irrelevant.

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Powers

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Logs

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Quaternion exponential multiplication

Andrew Millard suggested the result for the Grassman product.

Comments & Questions

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