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We know that during projection 3D space points $(x, y, z)$ projects to projection plane which has 2D points $(x, y).$ But during matrix calculation we use homogenous coordinates is of the form $(x, y, 1).$ And we know that projective plane is of the form $(x, y, 1).$

My question is that homogenous coordinates and projective plane points both are same thing, I mean "is all projective plane points are said homogenous coordinates"?

My second question is what is the difference between projection plane and projective plane? I mean "is projective plane is real plane or just imagination plane"? And we know that projection plane is real thing where we do all projection. But both have similar names. What is the relationship between these?

S. M.
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  • We use a 4x4 matrix for 3D transformations because a 3x3 matrix cannot encode 3D translation, but by augmenting the matrix we can make a 3D translation in the same operation. Since we move to 4D, we expand $(x, y, z)$ to $(x, y, z, w)$ where $w$ is $1$ for points (their position matters, so we put $1$ so they are translated), and $0$ for vectors (they don't have intrinsic position). Then we need to project back to 3D, and from there to 2D. See The view matrix finally explained. – Theraot Dec 19 '21 at 20:38
  • @Theraot I understand this as you said. But my question isn't that you said. I asking projective plane points and homogenous coordinates both are same thing or not?projection plane and projective plane both are not same. – S. M. Dec 19 '21 at 20:42
  • Projective plane is a related but separate concept. Looks like you just got an answer. – Theraot Dec 19 '21 at 20:43
  • @Theraot what is the difference/relationship between projection plane and projective plane? – S. M. Dec 19 '21 at 20:52

2 Answers2

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Short answer: the two concepts “projective plane” and “projection plane” are different things, though they are loosely related.

Longer answer …

The “projective plane”, often denoted by $P^2$, is an abstract mathematical concept. It’s used in a field of mathematics called “projective geometry”. As the other answer explained, the basic idea is to represent each 2D point by a 3D line passing through the origin. The benefit is that this allows you to represent 2D points that are “at infinity”. You can use this technique with any plane.

The “projection plane” is a specific plane that’s used in 3D computer graphics. The points of a 3D object are projected onto the projection plane to produce a 2D image. Quite often, the projection plane has equation $z=1$ in some coordinate system.

People often use 4D (homogeneous) coordinates and $4\times 4$ matrices to represent the 3D-to-2D projection in computer graphics. This approach is not much related to the projective plane $P^2$, but it is somewhat related to projective 3-space, $P^3$.

Similarly, if you use 3D (homogeneous) coordinates to represent points in any plane, you are effectively working with the projective plane, $P^2$. But note that this is true of any plane. In particular, it’s true of the projection plane that you use in computer graphics, so this is the connection between “projection plane” and “projective plane”.

The main reason homogeneous coordinates are used in computer graphics is so that perspective projection can be represented by a matrix multiplication. But you don’t have to use matrices and homogeneous coordinates if you don’t want to —- the whole projection calculation can be done just using ordinary 3D coordinates. And this approach doesn’t involve $P^2$ or $P^3$ or any other concept from projective geometry.

bubba
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  • when we use 2D transformation then we use homogenous coordinates $(x, y, 1)$ which is point of $P^2$ then in this case projection plane and projective plane isn't related with each other? – S. M. Dec 20 '21 at 06:50
  • They aren't unrelated concepts. 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). – Acccumulation Dec 20 '21 at 07:27
  • @Acccumulation could you answer this concepts with example? – S. M. Dec 20 '21 at 10:42
  • @Acccumulation. True. If you use 3D homogeneous coordinates in a plane, then in some sense you’re working with $P^2$. I’ll add this to my answer. – bubba Dec 20 '21 at 11:25
  • @bubba when projection plane at $z=1$ then properties of projection plane is same with projective plane? Or projection plane at $z=1$ then projection plane becomes a projective plane? – S. M. Dec 20 '21 at 13:24
  • @Acccumulation "those points exist in the projection despite not being in the original plane"-- you mean points at infinity $(x, y, 0)$ exists during projection but original projection plane doesn't contain $(x, y, 0)$? – S. M. Dec 20 '21 at 13:52
  • If you choose to represent points on your projection plane by 3D homogeneous coordinates, then you are (roughly speaking) treating it as the projective plane. – bubba Dec 20 '21 at 23:32
  • I’m guessing that you’re studying computer graphics. If so, I’d recommend that you focus on projection planes, forget about the projective plane, and stop worrying about the fact that they sound similar. – bubba Dec 20 '21 at 23:34
  • It doesn’t really make sense to talk about a projective plane. It’s an abstract mathematical concept, and there’s only one of them. That’s why people say the projective plane. – bubba Dec 20 '21 at 23:39
  • @bubba understood all the things. You just tell me, projective plane is just ordinary 2d flat plane like any other plane (for example projection plane) or projective plane is just just mathematical concepts? What is right? – S. M. Dec 21 '21 at 07:30
  • I’m sorry, but I don’t have any explanations that are clearer than the ones I already gave. Again, I suggest that you just ignore the projective plane, if it confuses you. You don’t need to understand the projective plane to do computer graphics. – bubba Dec 22 '21 at 12:16
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Each point of the projective plane $P^2$ can be represented in the form $[x : y : z] \in P^2$ for some point $(x,y,z) \ne (0,0,0)$ in $\mathbb R^3$. Using this representation we have $[x : y : z]=[rx : ry : rz]$ for any $r \ne 0$ in $\mathbb R$.

The ordered triple $(x,y,z) \in \mathbb R^3$ is called "homogeneous coordinates" for the point $[x : y : z] \in P^2$. But $[x:y:z]$ and $(x,y,z)$ are not equal. When a point $[x : y : z]$ of the projective plane is represented in homogeneous coordinates as $(x,y,z)$, that representation is not unique, and the point $[x : y : z]$ in the projective plane is a different mathematical object than any ordered triple $(x,y,z) \in \mathbb R^3$ that represents it in homogeneous coordinates.

Homogenous coordinates of the special form $(x,y,1)$ may not be used for the entire projective plane. They may only be used for a limited portion of the projective plane as follows:

  • A point $[x:y:z]$ in the projective plane such that $z \ne 0$ may be represented, using $r=\frac{1}{z}$, as $$[x:y:z] = [x/z:y/z:z/z]=[x':y':1]\quad\text{where}\quad x'=x/z, \quad y'=y/z $$

If you want to cover the entire projective plane, then the usual convention is to use two other special types of homogenous coordinates:

  • Points $[x:y:z]$ in the projective plane such that $y \ne 0$ may be represented, using $r=\frac{1}{y}$, as $$[x:y:z] = [x/y:y/y:z/y]=[x':1:z']\quad\text{where}\quad x'=x/y, \quad z'=z/y $$

  • Points $[x:y:z]$ in the projective plane such that $x \ne 0$ may be represented, using $r=\frac{1}{x}$, as $$[x:y:z] = [x/x:y/x:z/x]=[1:y':z']\quad\text{where}\quad y'=y/x, \quad z'=z/x $$

I'll add one more thing: given a point $(x,y,z) \ne (0,0,0)$ in $\mathbb R^3$, the formal definition of the projective plane tells you exactly what the point $[x : y : z] \in P^2$ is that is represented in homogeneous coordinates as $(x,y,z)$. This lets you see for yourself how, exactly, $[x : y : z]$ and $(x,y,z)$ are different.

Namely: $$[x : y : z ] = \{(rx,ry,rz) \in \mathbb R^3 \mid r \in \mathbb R\} $$ In other words, given $(x,y,z) \in \mathbb R^3$, the corresponding point $[x : y : z] \in P^2$ is identified with the line in $\mathbb R^3$ that passes through $(0,0,0)$ and $(x,y,z)$.

Lee Mosher
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