I haven’t really had time to read this material in detail. But I read enough to suspect that it’s worth looking at the claim that the dual construction used in euclidean PGA is anti-intuitive. This argument appears to be based on the perception that having the planes be 1-vectors means that you start with the higher-dimensional subspaces and build up the lower dimensional ones. I suspect however this argument is based on an incomplete understanding of duality.
I’ve ended up writing a somewhat long post to explain what I mean. Apologies to the speed freaks out there.
Why do we say that a point is 0-dimensional? It’s equivalent to saying that it’s indivisible, elemental, simple: It’s the geometric primitive I start with (in the standard construction) and out of which I incrementally build up all the other primitives by wedging. For example the wedge m := P_1 \wedge P_2 of two points in the standard construction is a 2-vector representing the joining line of the two points. The line is 1-dimensional since an arbitrary point P incident with m (that is, satisfying m \wedge P = 0) can be represented by the points P_1 + \beta P_2 for \beta \in \mathbb{R} (to include P_2 in this expression you can use homogeneous coordinate \alpha : \beta in \alpha P_1 + \beta P_2 but I don’t want to overly complicate things here). The line m in this sense is conceived of as consisting of all the points incident with it, it’s called a point range in projective geometry. Similar remarks apply to a plane p = P_1 \wedge P_2 \wedge P_3 as a 2-dimensional set of points, called a point field in projective geometry. To sum up: the dimension of a geometric primitive in the standard construction measures how many linearly independent points you need to generate the primitive.
What does this look like in the dual construction? Now we build up all the other primitives out of planes. A plane in the dual construction is just as simple and indivisible as a point is in the standard construction. We’re used to saying that the wedge of two planes m = p_1 \wedge p_2 is the intersection line of the two planes. But we can be, and need to to be, more precise. A plane p is incident with this plane if it satisfies m \wedge p = 0. This set of such incident planes is called a plane pencil and is analogous to the point range defined above. If we’re going to allow saying “the joining line consists of all the points incident with it” then we have to also allow saying now that “the intersection line consists of all the planes incident with it.” This pencil can be parametrized (exactly as above) as p_1 + \beta p_2 for \beta \in \mathbb{R}, so it’s 1-dimensional (of course with respect to the 0-dimensional building block, the plane).
Similar remarks apply to the wedge of 3 planes to produce a point: P = p_1 \wedge p_2 \wedge p_3. The set of planes p incident to this point satisfy p \wedge P = 0. Just as the plane in the standard construction can be thought of as all the points incident to it, the point in the dual construction can be thought of as consisting of all the planes that are incident with it. This set of planes is dual to the point field above and is called a plane bundle in projective geometry.
To sum up: once you’ve understood that
- dimension isn’t inherent in the geometric primitive but depends on the way that space is conceptualized, and
- that the projective geometric principle of duality provides a logically rigorous and historically proven basis for an alternative conceptualization of space,
then, based on my experience, the conviction that planes don’t make good 1-vectors can be overcome.
(Note that one still has access in PGA to both aspects of the geometric primitive – the Poincare duality map allows you to move over from the plane-based to the point-based Grassmann algebra and vice-versa. In fact that’s how the join operator (regressive product) is implemented. Example: you can’t join two bundles you have to join two (naked) points. We’re not committed unilaterally to the dual construction for everything, only insofar as it mirrors the metric relationships of euclidean space.)
One final comment: I don’t want to minimize the seriousness of the mental revolution required to think of points and planes in the dual way that I’ve described above. We are hard-wired at some level to prefer the “reality is build out of points” point of view (excuse the pun, it’s plane silly) to the newer alternative “reality is built out of planes”. It’s one thing to say that mathematically the two statements are equivalent but it’s something else to inwardly experience the equivalence as reasonable or believable.
Fortunately, using euclidean PGA successfully does not require that you subscribe to any view about how reality is built up. An educated skepticism is always justified. My recommendation: first make sure that the results are mathematically impeccable (I hope I have made a start with this post), then implement, apply to reality, check for discrepancies, and repeat. I’ve been doing that for 10 years now with PGA and haven’t found anything that doesn’t agree 100% with everything else I know about the world.
Also my article “Geometric algebras for euclidean geometry” especially Section 5, “Homogenous models using a degenerate metric” has some things to say that may be of interest to readers of this thread.