Questions about planes

[qhdqhd]

Thanks for your great work！
I'm doing exploratory experiments for my dissertation.
For static scenes, I have some confusion about the chosen three planes.
In the paper, three mutually orthogonal planes of x=0, y=0 and z=0 are used, and good results are obtained.
I made a linear weighted combination of the xyz coordinate axes to get three planes that are not orthogonal to each other, such as these three planes: x+y+z=0, x+y=0, y=0. Experimental findings seem to still achieve good results.
However, when I do non-linear combinations (such as multiplication and division) of xyz coordinate axes, such as x/z, I find that the effect becomes bad, with blurring and artifacts appear. I would like to ask what is going on here? Do you have an idea?
Thank you again.

[sarafridov]

I can make a guess as to what is happening here, but I haven't tried these sort of nonlinear plane combinations before. My guess is that maybe the difference is whether or not the original (x,y,z) coordinates can be recovered from the projected coordinates. If you choose 3 linear and independent combinations of dimensions for the three planes, then the three projections should give you enough information to solve for the original coordinates (3 linearly independent linear equations in three variables should have a unique solution). If the three planes are instead based on nonlinear combinations of dimensions, then depending on the specific combinations you may or may not still be able to recover the original coordinates based on the "projections" into the three planes. In your example of x/z, there is a degree of freedom that I could scale both x and z by the same amount and land in the same place in the x/z plane--if the other two planes contain enough information to distinguish this extra scaling factor then it should be fine, but if not then I'd expect the model to end up with streaky artifacts that blur along this extra degree of freedom.