I am studying Projective Geometry in 3D for Computer Vision. I am confused on the high-level rationale behind our need to map from heterogeneous to homogeneous coordinates, and I would like to confirm whether my understanding is correct.
Projecting a 3D Cartesian coordinate to a 2D plane in the Euclidean space is not a linear mapping. This is not ideal, as we can convert a 3D coordinate to its 2D projection, but not the reverse. Therefore, we define a 3D Cartesian coordinate (heterogeneous?) as a 4D homogeneous coordinate by increasing its dimensionality and multiplying by a constant. So, $(x, y, z) \rightarrow [X, Y, Z, 1]^T$. In the homogeneous space, we are able to define a linear mapping between a 4D homogenous coordinate and a 3D homogeneous coordinate. Then, by mapping the 3D homogeneous coordinate into a 2D Cartesian coordinate, we are effectively able to linearly map a 3D Cartesian coordinate into its 2D projection and the reverse.
If my understanding is correct, Cartesian coordinates are mapped to corresponding homogeneous coordinates to allow a linear mapping between 3D points and their projections. Is this correct? Thank you