# $P^2$ vs projection plane

I want to explain what I understood of definition of the two things.

Projection plane: The general processing steps for modeling and converting a world coordinate description of a scene to device coordinates, we need projection plane.

Projective plane: We know that $$P^2$$ is all $$\mathbb R^2$$ points and point at infinity.In projection plane any point exists in $$(∞,∞)$$ if we want to represents it then we need projective plane.

For instance, a point in Cartesian $$(1, 2)$$ becomes $$(1, 2, 1)$$ in Homogeneous. If a point, $$(1, 2)$$, moves toward infinity, it becomes $$(∞,∞)$$ in Cartesian coordinates. And it becomes $$(1, 2, 0)$$ in Homogeneous coordinates, because of $$(1/0, 2/0) ≈ (∞,∞).$$ Notice that we can express the point at infinity without using $$"∞".$$

My question is what's difference between Projection plane and Projective plane?

In computer graphics the projection plane is most commonly defined as a plane perpendicular to the camera at a specific distance from the camera (the distance is often labeled $$g$$). It is the plane that an image will be projected onto and is usually shown between the near and far planes of a projection matrix. In the attached image (from the fged website) it is the blue plane.

The projective plane on the other hand is usually shown as being parallel with the camera (or nearly parallel) extending out to infinity and typically has two lines draw on it like railroad tracks that meet (at infinity). Projective planes are not generally computed, though sometimes the need comes up. While a "projection plane" is mostly just conceptual, real world examples of a "projective plane" can be found. I attached a couple images showing some examples.

There are other definitions of projective plane (such as the one over at wolfram math) but from my experience these are the two most common definitions and what folks generally are referring to when using the terms.

• in the first image I can capture image by camera is green plane blue plane?
– user17337
Oct 11, 2021 at 13:32
• The first image is an example projection, the green plane is the near plane, the gray plane is the far plane and the camera would be positioned at $o$ which is the origin. Oct 11, 2021 at 14:01
• But which plane's image camera got captured, it is blue plane?
– user17337
Oct 11, 2021 at 14:02
• Yes, the blue plane. Oct 11, 2021 at 14:03
• "Projective planes are not generally computed" - don't understand this line... Please elaborate little bit.
– user17337
Oct 11, 2021 at 14:05