| ▲ | kevinventullo 4 days ago | ||||||||||||||||||||||
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 | |||||||||||||||||||||||
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