I try to wrap my head around intuitive definitions I can use (as a beginner) for 1, e0, e12 and e012 in PGA 2D… This would be my take:
For Plane-Based PGA 2D:
1: scalar “1” or entire vector space ≈ dual(e012)
e0: line at infinity ≈ dual(e12)
e12: origin (point) ≈ dual(e0)
e012: null vector space ≈ dual(1)
For Point-Based PGA 2D:
1: scalar “1” or null vector space ≈ dual(e012)
e0: point at infinity ≈ dual(e12)
e12: line at infinity ≈ dual(e0)
e012: entire space ≈ dual(1)
Am I completely off ?
in Point-based PGA, e0 is the point at the origin; that algebra is basically the algebra of homogeneous coordinates (HC) of points, and e0 = [0,0,1] is the point at (0,0). Similarly, e1 =[1,0,0] is the point at infinity (in the e1-direction), you would call it a direction vector when doing HC
Thank you @LeoDorst for your enlightenment about e0 and e1
I currently picture the 2D Projective GA with the following artifacts:
- The infinity as an embedding Sphere representing the Space (Universe).
- The Source as the point at [0, 0, 0] (center of the Sphere).
- The projective plane “P” at [1, 1, 1], as a section of the infinity Sphere (so a disc) at “z” 1 (if we consider going “up” is along “z” inside the Sphere).
- Then, a line blasting from the Source [0, 0, 0] will cross the plan “P” as a point [x, y, 1] and hit the Sphere (infinity) as a direction.
- Then, a plane crossing the Sphere will materialize on “P” as a line and cross the Sphere (infinity) as an horizon (a line on the surface of the Sphere, so a circle).
I don’t know if it makes sense. But I derived this vision from several materials I could see online from you or @enki. Especially when you touch the Dual question and illustrate it.
I’m very appreciative of the Geometers () community.