Regarding the projective space

Hello All!

I am reading this excellent collection of articles on projective spaces, transformations etc.

I came across this sentence:

Note that in projective space points (meaning location) and translations are represented by different types (unlike vector algebra) this is reasonable as they are different entities.


I’m not able to understand what exactly that meant.
PS: In projective space, using homogenous coordinates, we get
(x, y, z, w) = point if w != 0
and (x, y, z, w) = vector if w = 0, for a 4D projective space.
Am i right?

So, what exactly does the author try to explain here?

Thanks in advance!

(x,y,z,w) with w = 0 is also a point (provided one of x,y,z != 0). These are the “infinite far” points of the projective space, so you could say that they represent directions, but a translation is defined by a direction plus a length or distance.

A translation in a projective space would be represented by a translation matrix.

Thanks for the reply mbentrup!