What is the relationship between projection plane and projective plane?

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.

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 $(x,y,z)$.