Indeed, projective GA does not at all require a degenerate metric, which means the Hodge complement is the most natural to use. The only time the non-metric complement might be useful is with a degenerate metric, which projective GA does not require. As discussed by Hestenes, the mother algebra of R(n,n) contains all possible projective splits, without using a degenerate metric.
Perhaps you are in the wrong forum, because all of projective GA is contained in Hestenes mother algebra, which does not have a degenerate metric.
In non-degenerate metrics, the Hodge complement is the most natural for geometric algebra, since it is founded in the unmodified geometric product.
I should have linked to the “computation of the hodge star” section.
I the context of bivector.net’s “GA for CGI”, I’m going to take the CGI definition of the Hodge Star. Now the definitions used in the Computer Graphics community are not always correct, but after that first paper is written and referenced enough times it becomes “truth”.
“More generally, suppose e1, . . . ,en is an orthonormal basis for Rn. If we start out with k
orthonormal vectors u1, . . . , uk, then the Hodge star is uniquely determined by the relationship
(u1 ∧ · · · ∧ uk) ∧ *(u1 ∧ · · · ∧ uk) = e1 ∧ · · · ∧ en.
In short: if we wedge together a k-dimensional “unit volume” with the complementary (n − k)-
dimensional unit volume,” we should get the one and only n-dimensional unit volume on Rn.”
To me this seems compatible with a degenerate metric given a trivial determinant ( orthonormal basis). When you use the resulting k-vector in most situations, you will be doing so through applying k-forms, where your basis 1-form application for the degenerate 1-vector and it’s 1-form will be 1. This again seems ok to me.
You’ve been referencing your paper quite often. Not many are able to read it. Would you be able to write an easier to read and understand version of the theory in your paper? Like a tutorial blog post?
Apologies if my tone is a bit harsh, I’m just only trying to defend the Hodge complement, which I think is an important concept for the geometric product theory.
Yes, I intend to write a more detailed version of my notes with examples and more background. However, it may take a while before that’s available.
Note that Keenan Crane’s definition of the Hodge Star begins with an orthonormal basis of \mathbf{R}^n. So there is a metric underlying the definition.
But the dependence on the metric is not clearly stated. For example let u = e_1 \wedge e_2. Then *u := e_3 satisfies u \wedge *u = \bf{I} the unit pseudoscalar. But so does *u := \alpha e_1 + \beta e_2 + e_3 for any \alpha and \beta doesn’t it? (Since e_1 \wedge e_2 \wedge e_1 = 0, etc.) That to me implies that there is no well-defined “complementary” (n-k)-volume. That only comes with a metric that defines what orthogonal means. In this case, that is presumably the (n,0,0) positive definite metric of \mathbb{R}^n. But the definition then needs to make that dependence clearer it seems to me. The definition of Hodge star in the Wikipedia article makes this explicit.
So it seems to me that the Hodge star operator is just a disguised form of multiplication by the pseudoscalar. It’s not phrased that way since it’s defined on the exterior algebra where multiplication (wedging) with the pseudoscalar gives 0, but the result is the same since both produce the orthogonal complement of the given k-vector.
A general note about terminology (based on the projective geometric roots, with apologies if I’m repeating myself):
I recommend referring to multiplication by the pseudoscalar as the polarity on the metric quadric or metric polarity or absolute polarity for short.
The map G \rightarrow G* taking an element in one exterior algebra to the same element in the dual exterior algebra I recommend referring to as the dual coordinate map or duality for short. It shares many properties with an identity map.
@Orbots: The reason that 1. and 2. appear so similar is that when you write the maps out in coordinate form, they appear identical. But due to the different target spaces they mean very different things.
Further discussion of this can be found in Sec. 5.2 of the article “Geometric Algebras for Euclidean Geometry” at https://arxiv.org/abs/1501.06511.