Were does the notation T^2 for oriented real projective space come from? That's just bad, because it is not a torus but a sphere, and the two are topologically very different!
If you click through the first link there’s an explanation:
“In precise mathematical terms, this set of rays is called the oriented real projective plane and is commonly denoted by T^2. If you’ve seen this terminology before, you’ll notice that this is a torus. This is because in real-projective geometry, we also add the points and lines “at infinity”.”
The oriented real projective plane is a sphere, not a torus.
The projective points at infinity (one point for every 1-D angle (R mod 2pi)) form the equator of the sphere.
The T in T² is for "two-sided" , not Torus.
The torus explanation that Tangram gives doesn't make sense.
In a pinhole projection, the horizontal and vertical infinites do not "wrap around" to meet. There is no meaningful "horizontal" and "vertical", the system is rotationally symmetric, which forms a hemisphere of curve it to make it compact. (Half sphere because you can only see one half of the space outside a pinhole camera)
Thanks for sharing such an insightful article of complex material. Some follow-up showing how this helps optimization (gradient descent, Newton solvers?) would be great.
>> For convenience of notation, we’ll drop the explicit parametrization of the curves and denote this vector shift as [...]
FYI for clarity I wish the explicit parametrization was kept, even though it is more verbose.
“In precise mathematical terms, this set of rays is called the oriented real projective plane and is commonly denoted by T^2. If you’ve seen this terminology before, you’ll notice that this is a torus. This is because in real-projective geometry, we also add the points and lines “at infinity”.”