> It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3
with a single linear operator on R^4.
I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces and their transformations by including "points at infinity"), contains no concepts of length or angle (and therefore no equivalent of translations and rotations) and is in some sense "geometry with only the straightedge, no compass".
Curious if I'm just missing something there, though. I'm no expert on any of this.
> It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4.
I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces and their transformations by including "points at infinity"), contains no concepts of length or angle (and therefore no equivalent of translations and rotations) and is in some sense "geometry with only the straightedge, no compass".
Curious if I'm just missing something there, though. I'm no expert on any of this.
[1] https://en.wikipedia.org/wiki/Projective_geometry