One thing that I've had trouble understanding is why homogenous coordinates are "scale ambiguous". If I have a homogenous coordinate (5, 10, 2), how is that scale ambiguous? Doesn't that tell me that they are scaled by 2?
For instance, I was reading some notes and I saw this:
"The equation is homogenous, i.e. f can only be determined up to scale".
What does that mean? "Can only be determined up to scale"? Thanks.
Homogenous coordinates for $\mathbb R^n$ are coordinates in $\mathbb R^{n+1}$ under the equivalence relation $(x_1,...,x_{n+1})\sim(y_1,...,y_{n+1})$ if there exists some non-zero $\lambda$ such that \begin{align}x_1=\lambda y_1\\x_2=\lambda y_2\\\vdots\\x_{n+1}=\lambda y_{n+1}\end{align}
Another way to visualize homogenous coordinates is that the projective space of dimension $n$ is the space of lines intersecting the origin in $\mathbb R^{n+1}$. This way of visualizing the space is nice for the projective space of dimension $2$ since we can visualize it by seeing how it projects onto a sphere, and onto a plane.
Source for the picture.
You can see how we can associate every point on the plane with a line through the origin (but not vice versa), and how we can associate every point on a hemisphere of the sphere with one of these lines. You can now see why scaling doesn't matter, since we still belong on the same line (or in the definition above, our equivalence relation is on scaling operations).