Source
- URL: https://x.com/EricLengyel/status/1826161505223717336
- Author: Eric Lengyel (@EricLengyel)
- Posted: 2024-08-21 07:36:45
Thread
1/ @EricLengyel
Following up with 3D homogeneous coordinates. What would you think if someone (not me) told you it was more geometrically intuitive for the vector on the left to represent the red line? And for the bivector on the right to represent the red point?
2/ @pedroth9
@EricLengyel
If you think of the Hodge star of the vector, then the vector turns to a bivector which represents a line. The bivector turns to a vector with the Hodge star which represents a point.
3/ @EricLengyel
@pedroth9
The Hodge star does not do what you think it does in this case due to the degenerate metric. But the complement does. You are correct that the red objects correspond to the subspace orthogonal to the one occupied by the blue vector or bivector.
4/ @EricLengyel
@pedroth9
(In this case, the Hodge dual of a line is the origin, and the Hodge dual of a point is a line through the origin parallel to the red line shown.)
5/ @gary_snethen
@EricLengyel @pedroth9
Isn’t the 3D dual of the blue vector the bivector that represents the red line? I think that is what @pedroth9 was suggesting.
6/ @EricLengyel
@gary_snethen @pedroth9
Yes, that’s what I meant by “You are correct that the red objects correspond to the subspace orthogonal to the one occupied by the blue vector or bivector.” They are related by duality, but algebraically they are complements. The Hodge dual isn’t quite the same thing.
7/ @EricLengyel
My point isn’t about whether the geometric representations make sense if you know what’s going on. They do. The question is, are they more intuitive in dual form.
8/ @gary_snethen
@EricLengyel @pedroth9
Clearly not. :)
Although perhaps some people might find the mathematical operations more natural after conversion.
But geometrically, your blue diagrams are universally more intuitive than the red.