Scaling in Homogenous coordinates

Question

I am reading Forsythe and Ponce (Computer Vision) and it says: "We will use the convention that homogeneous coordinates are only defined up to scale, whether they represent points or planes." I want to know what does "up to scale" means in the above statement. I'm a newbie to homogeneous coordinates, so please answer in simple English. I also saw this link https://stackoverflow.com/questions/17114880/up-to-a-scale-factor But I didn't get it, so if anyone can elaborate, that would be great.

Answer 1

One should think of projective space as "all lines through the origin.

So, in general, given a line $y= \lambda x$ (where $x \neq 0$), we think of these as a single point in space. In other words, we make the identification $(x,y) \sim(\lambda x, \lambda y)$, where $\lambda \neq 0$, since these two determine the same line (i.e the line is determined by its slope, so $y/x=\lambda y/\lambda x$) so in homogeneous co-ordinates,

$$[x,y]=[\lambda x, \lambda y],$$

for any scalar $\lambda \neq 0$.

The real point here is that the equivalence relation $\sim$ is one way to represent all lines through the plane. If you would really like to see this in a hands on way, pick a "representative" for the equivalence relation.

One way to do this is to select the line $x=1$ and each line that is not the line $x=0$ will intersect this vertical line in one unique place. Choose these points to be your representatives, and then it will become clear that any line that differs from another by a scalar $\lambda$ will have the same point as its representative.