This post is addressed to readers curious about the genesis of the “ideal norm” used in euclidean PGA and responsible for much of the striking polymorphicity of its API, and can be seen as a continuation of the discussion in the course notes (Sec. 7.1). I was motivated to share these remarks as I responded to a post on the connection between planes and their normal vectors (that are ideal points).
Recall that the ideal norm of an ideal point (x,y,z,0) is just the standard euclidean vector space norm \sqrt{x^2+y^2+z^2} (whereas the euclidean PGA point norm yields \sqrt{w^2}=0).
The ideal norm guarantees that ideal points behave exactly like euclidean vectors (i. e., elements of \mathbb{R}^n). Presented in this way, the ideal norm appears to be rather ad hoc. I hope to make clear here that there is a deeper connection.
The ideal norm in fact arises if we require that the angle between two ideal lines should be the same as the angle between any two euclidean planes whose intersection with the ideal plane is these two ideal lines. Considering the ideal plane to be a 2- (or in general an n-1) dimensional dual PGA defined by this “induced” inner product on its 1-vectors leads exactly to the ideal norm described above.
It’s probably better to present the “standard” norm from the start as restricted to the euclidean geometry (not including the ideal plane) and the ideal norm also from the start as the “consistent” complement of this norm on the ideal plane (consistent in the sense that angles between planes agrees with the angle between their ideal lines) rather than (as in the course notes) implying that the standard norm is being replaced by the ideal one. The Cayley-Klein construction after all only defines an inner product on the euclidean elements, not the ideal ones.
In fact, rather than starting with the euclidean planes and deducing the “induced” inner product on ideal lines as sketched above, I find it more mathematically appealing to start with this inner product on the ideal lines and “push” it onto the euclidean planes (i. e., the inner product of two planes is defined to be the inner product of their two ideal lines).
This approach to the ideal norm is sketched in the appendix of the paper “Doing euclidean plane geometry using projective geometric algebra”. available here. (This appendix in general shows how to develop PGA without using coordinates and assumes some familiarity with modern abstract algebra.)