Join and meet in projective geometric algebra

I’m posting this due to a thread on the bivector discord channel that raised the question of what exactly one should associate with the terms “join” and “meet” in PGA, projective geometric algebra. Since I’d like to be able to find this answer again, I’m posting it here and providing a link to this post back on discord.

The answer to this question of “join” and “meet” contains some surprises due to the existence of both plane-based and point-based PGA.

In any GA, the wedge (or “progressive”) product A^B of two (linearly independent) 1-vectors A and B consists of 1-vectors P satisfying P^A^B = 0 (so-called “outer product null space”). This is a 1D family of 1-vectors, the “span” of A and B.

Examples: In a point-based PGA, where 1-vectors are points, this yields the joining line between the two points as a set of points (“point range”). In a plane-based PGA, such as euclidean PGA, 1-vectors are planes and the wedge yields a set of planes that all pass through the common line of A and B (that’s called a “plane pencil”), the meeting line of the two planes. So the same wedge operation can appear as join or meet depending on whether you are in a point-based or a plane-based algebra. In both cases however, the word “span” would be accurate.

The following figure shows the two representations of a 3D line: as point range (on the right) and plane pencil (on the left). These are sometimes called spear (point range) and axis (plane pencil).

That’s not all. Every GA also has a regressive product that can adds another flip to the logic. The regressive product, unlike the span, gives the intersection of its arguments, considered as sets.

Example: In a 3D point-based PGA, 3-vectors are planes, each thought of as consisting of a 2D extent of points. The regressive product aVb of two planes, then, consists of the points common to both, that is, the meeting line (as a point range). Recall that the wedge (or progressive) product of the two planes in the plane-based algebra produced the same line, but as a set of planes (plane pencil). Something similar happens in a 3D plane-based PGA: 3-vectors are points, each considered to consist of all the planes that pass through the point (the so-called “plane bundle” of the point). Then you can recognize the set of planes common to both bundles as the “plane pencil” of the common line of the two points, considered this time as a plane pencil rather than as a point range. Just like “span” was a neutral term for the progressive product, a neutral term for the regressive product would be “intersection”.

Both the common line of two points and the common line of two planes can be obtained with either progressive or regressive product, and as either a “span” or an “intersection”. Care is advised when using the terms “join” and “meet” since their meaning depends on whether the context is point-based or plane-based.

This theme is part of the more encompassing topic of “geometric duality” whereby every geometric primitive appears in projective geometry twice, once in plane-based form and once in point-based form like the line in the example above that can be considered as both a point set (point range) or a plane set (plane pencil). An introduction to geometric duality in PGA can be found in Sec. 8 of the preprint “A bit better: Variants of duality in geometric algebras with degenerate metrics”, [2206.02459] A bit better: Variants of duality in geometric algebras with degenerate metrics.