3 ms·

What does the differential operator do and how does it relate to geometric algebra?

• ajkjk 7y ago
  There's a lot of connections between projective geometry and geometric algebra (well-- at least exterior algebra. Not sure about 'geometric', because I don't know what the geometric product means). If you implement _oriented_ projective geometry in homogenous coordinates (so a point (x,y) is a vector (x,y,1)), then the meet and join operators are implemented as ∧ and ∨. You can make a little dictionary: Join = ∧, Meet = ∨. Vector = point, Bivector = line, Trivector = area, etc. The figure spanned by points (a,b,c) = a ∧ b ∧ c. The boundary of the figure = ∧^(k-1) of the metric (a,b,c), equal to ∂(a,b,c) = a ∧ b + b ∧ c + c ∧ a. I have a very amateur blog that I never publicize about this stuff and I had a long post about this, but I've taken it down for now to rework it, or I'd link it here. Suffice to say there's a lot of connections and I feel like there are even more here that haven't been discovered yet. The book "Oriented Projective Geometry" by Stolfi has a lot of this, although it doesn't explicitly talk about geometric algebra or the wedge product -- but it uses all the same symbols. I'm on the lookout for a better reference that bridges the gap. [I have so far not figured out what the exterior derivative means in projective geometry, besides being dual to ∂; I believe that if derivative operators are just dual to basis vectors, then d is literally just dual to ∂. Not sure. I also have no idea what the geometric product means, and tend to be skeptical of it for that meaning.]