# 3D homogenous coordinates versus 4D homogenous coordinates

We know that any 2D point $$(x, y)$$which represents as 3D homogeneous coordinates is of the form $$(x, y, 1)$$ which is the points of projective plane $$P^2.$$

If I use the same concepts for 3D points $$(x, y, z)$$ which represents as 4D homogeneous coordinates is of the form $$(x, y, z, 1).$$

My question is $$(x, y, z, 1)$$ could be the points of 3D projective space?

My second question is always homogenous coordinates are the points of projective space ?

N. B. -- $$P^2$$ is projective plane which contains the points of $$\mathbb{R^2}$$($$\mathbb{R^2}$$ points in $$P^2$$ can be represents as $$(x, y, 1)$$ ) and ideal points, is of the form $$(x, y, 0)$$ which is called points at infinity.

• Is $P^2$ supposed to be the real projective plane $RP^2$? Commented Nov 22, 2021 at 0:26
• @lightxbulb explicitly $P^2=\mathbb{R^2}$+ ideal points. Commented Nov 22, 2021 at 3:14
• Is the $+$ supposed to be union? Define "ideal points". Better yet - post the reference that you are using which employs this $P^2$ notation. Commented Nov 22, 2021 at 6:32
• @lightxbulb ideal points is of form (x, y, 0) which is point at infinity. Commented Nov 22, 2021 at 7:01
• The medium article explains your confusion. They define the term projective plane to be the plane $(x,y,1)$. It is true by their definition. This is usually not how the term is used as it refers to the real projective plane which is not just $(x,y,1)$. I could not find the definition of the medium article's author anywhere else (including the book that they cite). But by their definition $(x,y,z,1)$ would correspond to their definition of projective plane in 3D. I advise against using this terminology as I believe it is nonstandard and confusing (conflicts with the original term). Commented Nov 22, 2021 at 10:00