4 ms·
What I find most amazing about homogeneous coordinates is the duality between points and lines and their behavior under vector cross product. Basically: cross
by Scea91 9y ago
What I find most amazing about homogeneous coordinates is the duality between points and lines and their behavior under vector cross product.
Basically:
cross(p1, p2) = line_between_p1_and_p2
cross(line1, line2) = point_of_intersection_of_lines_1_and_2
To understand this you would need to dive a bit into projective space and how lines are represented there, but it is very enlightening.
- user51442 9y agoAnd in 3 dimensions it's points and planes that are dual - and finding the line between two points or the common line of two planes can now be done in a similar way, but using Plücker coordinates for the lines instead.
- theoh 9y agoThose two operations are called "meet" and "join", and in principle they work with any linear subspaces (point, line, plane, etc.) So the meet of a line and a plane in 3D is a point, the join of a point and a line in 3D is a plane, etc. Wikipedia is good on this stuff if you are patient with mathematician's way of expressing things... https://en.m.wikipedia.org/wiki/Join_and_meet https://en.m.wikipedia.org/wiki/Join_and_meet https://en.m.wikipedia.org/wiki/Linear_subspace https://en.m.wikipedia.org/wiki/Linear_subspace "Lattice of subspaces The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation."
- tel 9y agoThen you can go full geometric algebra and base your whole geometry on this latticez