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I asked this question to understand difference between projection plane and projective plane. But below the @bubba's answer, the user @Acccumulation comments this text which given below:

The projection plane is where you're projecting onto a plane. The projective plane is where you're projecting from a plane, and filling in the points at infinity (those points exist in the projection despite not being in the original plane).

My first question is what does mean second sentence of this comments? I mean when we do projection from 3D space to 2D onto projection plane, but what he mean, he means 3D space exists in projective plane?

When see the answer of @Lee Mosher he said homogenous coordinates are the points of $\mathbb R^3.$ But we see in many other answers homogenous coordinates are the points of projective plane $P^2.$ My second question is what is right? Homogenous coordinates belongs to $\mathbb R^3 or P^2?$

S. M.
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  • Let me answer the last question, to get you started. $\mathbb RP^2$ is a quotient space of $\mathbb R^3$. In particular, that means that for homogenous coordinates to belong to $\mathbb RP^2$, they must belong to $\mathbb R^3$ and also respect the quotient operation. (In other words, you can represent homogenous coordinates in $\mathbb R^3$, but the space has no knowledge of the fact that you are treating them as homogenous. It is more correct to say they belong to $\mathbb RP^2$). – Rushabh Mehta Dec 20 '21 at 15:51
  • For the first question, the projective plane $\mathbb RP^2$ is actually canonically embedded in $\mathbb R^3$ (as I said in the previous comment). So, you can view it as a projection of the plane onto 3D space, and some points added in to compactify the resulting projection. The projection plane is in the opposite direction. – Rushabh Mehta Dec 20 '21 at 15:53
  • @Don your second comments don't understand.. – S. M. Dec 20 '21 at 15:57
  • What part do you not understand... – Rushabh Mehta Dec 20 '21 at 15:59
  • @Don these two sentences "the projective plane RP2 is actually canonically embedded in R3 "."The projection plane is in the opposite direction" – S. M. Dec 20 '21 at 16:01
  • The first one, you should understand from Lee Mosher's answer to your other question. The second is obvious: the projection plane sends 3D space to 2D space. – Rushabh Mehta Dec 20 '21 at 16:02
  • @Don please explain easily what does the mean quotient space RP^2 and what is quotient operation with one little example. – S. M. Dec 20 '21 at 16:09
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    I won't. Not only did Lee do this in his answer, but it is literally one google search away as well. – Rushabh Mehta Dec 20 '21 at 16:10
  • I don't think "projection plane" is a standard term. That said, $\Bbb RP^2$ is the space of all lines in 3D through the origin. That means quotienting $\Bbb R^3\setminus{(0,0,0)}$ by the equivalence relation $\vec{x}\sim\lambda\vec{x}$. Since every line goes through the unit sphere, this is equivalent to quotienting $S^2$ by the relation $\vec{x}\sim-\vec{x}$, i.e. identifying antipodal points. Equivalently, since every pair of antipodal points has a one point in a given hemisphere, we can quotient a disk by identifying antipodal boundary points. – anon Dec 20 '21 at 17:35
  • This is all standard stuff. Google "projective plane RP^2" and you will find mountains of resources for you to choose from to help you understand. You can also look up stuff like quotient topology and equivalence relations. (Also I feel compelled to point out the statements "[RP^2 is] canonically embedded in R^3" and "RP^2 is a quotient space of R^3" are false statements.) – anon Dec 20 '21 at 17:37

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