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▲ red_trumpet 4 days ago

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!

▲kevinventullo 4 days ago | parent [-]

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”.”

▲lupire 4 days ago | parent [-]

It's incorrect, though.

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)

https://en.m.wikipedia.org/wiki/Oriented_projective_geometry

▲ChickenSando 4 days ago | parent | next [-]

You're right. I've messed up. The notation here is a bit misleading and I didn't dig as deep here as I should have. I'll fix the website soon.

▲erwincoumans 3 days ago | parent [-]

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.

▲kevinventullo 4 days ago | parent | prev [-]

Indeed, I should have been a more critical reader!