Relationship between Euclidean lines and projective points I'm learning the very basics of projective geometry, and I read the following in Stan Birchfield's notes, An Introduction to Projective Geometry:

Why is it that "overall scaling is unimportant"? In his notes, Birchfield uses this to build the idea that what we think of as points in Euclidean geometry can be thought of as lines in projective geometry?
 A: Observe that it is  a point in the affine plane (dimension 2) which corresponds to a line in a vector space  of dimension 3. The points  of a projective plane are just vector lines in this vector space.
As a vector line  is defined by a single vector, which be chosen as you like, provided it is non-zero, it explains  why ‘the (non-zero) scaling is unimportant’. So a point in the projective plane is defined by the ratios of its three coordinates, $(X:Y:T)$, for which not all three coordinates are $0$. If $t\ne 0$, one gets back a points of the affine plane with the map $\;(X:Y:T) \longmapsto \Bigl(x=\dfrac XT, y=\dfrac YT\Bigr)$. Conversely, a point  $(x,,y)$ in the affine plane corresponds to the points $(x:y:1)$  in the projective plane. The point with projective coordinates $(x:y:0)$ are considered as the ‘point at infinity’ in the direction $(x,y)$.
A: Informally, consider the data that your eye receives: light enters from a certain direction and registers information about what you see in that direction. You can tell how far away something is through various cues (such as the information from your other eye, relative sizes of known objects, overlapping shapes, etc.), but strictly speaking, everything along that line of sight is being collapsed into a single point.
This is what makes it possible to look at a cell phone and recognize that the camera is seeing what you are seeing, so to speak, even though the screen is a flat surface whereas you are seeing in three dimensions (or at least that's what your brain is telling you).
