I don't get how people are using homogeneous coordinates in order to construct the projection of an object. I know that homogeneous coordinates allow us to perform affine transformations in higher dimensions to make non-linear transformations linear. Is it the only purpose of homogeneous coordinates?
Consider the following picture
From the picture we can construct projection formulas in order to get projected points:
It is obvious that we can obtain $x_i$ and $y_i$, which are projected coordinates, just using simple triangle proportion calculations. This method does not require extra coordinate stuff unless we are interested in projection matrix construction which is non-linear transformation.
So, the main questions here are:
- What is the main purpose of homogeneous coordinates apart making non-linear transformations linear?
- Why we need points/lines at infinity if we can simply represent them as regular points/lines in order to obtain object projection (notice that I haven't used the concept of point's at infinity in order to obtain object projection)?
- Why homogeneous coordinates are called projective coordinates if they just extend dimension and that's it?
Can you, please, point me where I'm wrong and where should I go further in order to get better understanding of what is projection and how homogeneous coordinates are used in pure projection geometry? Maybe give some problems that show how homogeneous coordinates are really used?
PS: I've red tons of topics here and in different books, but left unsatisfied with the answers