Let x,y,z be points taken exclusively from the positive orthant.
For the scaling transformation
x'=x/(x+y+z)
y'=y/(x+y+z)
z'=z/(x+y+z)
where each function is a linear fractional transform
how can I interpret this?
That is, does this sort of componentwise LFT enjoy all the usual LFT properties?
I am trying to figure out if this transformation of mine preserves the cross ratio
and am hoping to use the properties of an LFT to do so.
Is this going to work out like I want it to?
I am having no luck finding anything relevant in the literature. Any suggestions there?



As mentioned, this is a central projection to the $x+y+z=1$ plane, or, in the positive orthant, you can also view it as an $L_1$norm normalization. Preservation of circles:
Cross ratios:
If you're interested in some other kind of crossratio, you have to define how you intend to "multiply" and "divide" the differences of points. Edit: See corrections. 

