In my computer vision course, we apply homogenous coordinates to represent points in 2D or 3D space. However, in my course it lacks a proper definition of what homogeneous coordinates are.
I have found according to Wikipedia that they are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.
Therefore, what would be a proper definition of homogenous coordinates in geometrical formation of images because I don't understand how much projective geometry differs from cartesian coordinates.
Have a look at the following figure:
It explains that the fondamental property of homogeneous coordinates to be "up to a factor" here
$$(x,y,z) \equiv \left(\frac{x}{z},\frac{y}{z},\frac{z}{z}=1\right) $$
is in correspondence with the fundamental process of vision : an observer whose eye is situated at the origin of 3D coordinates sees a 3D object (here a cube) through rays that intersect a kind of "screen" situated at $z=1$ (in fact this screen could placed at any height $z=z_0>0$) ; the 2D resulting image of the edges is the 2D rendering of the real 3D object...
This graphical example shows that in fact,
a point in 2D projective geometry is a line issued from the origin in 3D space with homogeneous coordinates,
in the same way, a line in 2D proj. geom. is a plane passing through the origin in 3D space, etc.