biVector forum

Reference papers for the 'dual' (e^2=0) maths

In the Siggraph 2019 lecture ( it is mentioned that the dual space representation which adds the e^2=0 is a recent discovery.

Do folks have any references to the relevant papers or articles on how that duality is considered?

I’m particularly interested in how it compares with physics duality (e.g. twin slits, Fourier duality, and fractional Fourier transforms)

The ability to pin down duality as a mathematical concept / technique looks very helpful.

(Electro-Optics Systems Engineer Retd)

Hi @Philip_Oakley,
I’m not sure from the wording of your question if I understand it.
The “dual space representation” and the e_0^2=0 condition are independent of each other; together they yield a geometric algebra for euclidean space.
Dual space representation
In general, the duality that arises in PGA (projective geometric algebra) is the duality of projective geometry. Duality is a deep symmetry of projective geometry in which geometric primitives and operations are paired with dual partners so that the dual of any true statement is also true. Dualization is a mechanical translation process that replaces terms in the original formulation with their dual terms.
For example, in n=3, duality is between points and planes, between lines constructed considered as the join of points (spears) and lines considered as the meet of planes (axes), between the join and meet operators, etc., etc.
It’s possible to define dual exterior algebras and dual geometric algebras by dualizing the standard definitions of these algebras. Hence, in the dual algebra, 1-vectors are planes, and the wedge operator \wedge is the meet operator; the regressive product \vee is then the join operator. More on this in the SIGGRAPH course notes in Sec. 5.10.
This topic is also handled in textbooks on multilinear algebra, for example, Greub, Multilinear Algebra, Springer, 1967.
Degenerate metric e_0^2=0
Any metric can be applied to either a standard or a dual algebra. In particular, Jon Selig showed in 2000 (see his book Robotics, Springer, 2006 for exact citation) that you can model euclidean space using a dual algebra combined with a degenerate metric satisfying e_0^2=0, otherwise e_i^2=1. A degenerate metric can also be combined with a standard algebra, but it doesn’t produce a model for euclidean space, but rather a model for polar euclidean space. (For details of polar euclidean space see for example my thesis, Ch. 10).


Thank you for the comprehensive reply. I will have a look at those references.

Part of the broader problem is in the way ‘duality’ is used so widely, and confusingly, in the different parts of science and maths (e.g. wave - particle, …). It’s not always clear in those different cases how the ‘duality’ manages to pin down both sides of the dual viewpoints.

I thought I’d heard within the Siggraph video the implication that the addition of the e_0^2=0 term was an important part in the full-up PGA, but I may have misheard.

When you mention the quaternions and dual quaternions, I presume you are referring to the distinction between the ‘vector orientation’ and the ‘plane orientation’ as the two duals. I hadn’t heard the term ‘dual quaternions’ before, so wasn’t exactly sure about the distinction.

I have worked a little with quaternions on inertial navigation, and looked at Maxwell’s equations in quaternion form (which reduces the number of equations), so they continue to be an interest.

Perhaps it’s the term ‘degenerate’ (as per the metric) that I was really looking for. I’ll have a look at the thesis, as I’m especially interested in how Units and Dimensions should be carried through SI calculations, especially when they loose the all important ‘radians’ (which are needed often in optical system design)