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Non-metric alternative to reciprocal frame

Much of the standard literature of geometric algebras assumes that the metric is non-degenerate. Several common computational techniques are based on this assumption. Such techniques can not be applied for degenerate metrics (such as the (3,0,1) metric of euclidean PGA).

This post addresses the use of reciprocal frames, which is one of these techniques.

In an (non-degenerate) inner product space V with an orthonormal basis \bf{e}_i, the coordinates b_i of an arbitrary vector \bf{b} can be calculated as b_i = \bf{b} \cdot \bf{e}_i.

If the basis is not orthonormal, this formula no longer holds. It is standard practice to introduce a reciprocal basis \bf{r}_i that satisfies b_i =\bf{b} \cdot \bf{r}_i.

This reciprocal basis however does not exist when the inner product is degenerate.

It is then advisable to ignore the inner product. This is actually no problem; after all, a metric is not needed to calculate the coordinates of a vector in a vector space.

Instead, apply the canonical dual basis \bf{e}^i of the dual vector space V^* to obtain b_i = \| \bf{b} \wedge \bf{e}^i \|, where the right-hand side is the magnitude of the pseudoscalar. This pseudoscalar essentially measures the “distance” of \bf{b} to the plane \bf{e}^i – up to the choice of a unit distance for each coordinate (remember, we’re not actually using the metric directly here).

Details describing what the canonical dual basis is and how to calculate it in the geometric algebra are available in this 3-page note here.

Questions and feedback welcome.

3 Likes

It should be possible to use dollar signs $ just like in \LaTeX to do inline \LaTeX typesetting, e.g. $\LaTeX$.

I’m looking forward to reading your article on the train later today :).

Thanks, @tBuLi, ... worked just as advertised.