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Is the idea of homogeneous coordinates only work in 2D and 3D?


I'm still a little confused on how any point on L can be used to describe p. Why isn't it just p only?


Is there ever any disadvantage to using homogeneous coordinates? It seems like you only need to augment a few values to your original matrices to represent them, so just wondering if there's ever a reason to not use them for these kinds of applications.


Is there any advantage to using homogeneous coordinates in higher dimensions---i.e. turning a 2D point into a plane in 4D? Does this let us represent additional transformations?


Are there any situations where using homogeneous coordinates is not worth it? I.e. the memory constraints of having to store an extra dimension or computation costs


When the dimensions get higher, it is still a good idea to use homogeneous coordinates?