Question

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?

Answer 1

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:

• Circles in planes parallel to $x+y+z=1$ are preserved.
• Circles (and any other shapes) in planes normal to $x+y+z=1$ passing through the origin are projected to line segments.
• Other circles are projected to ellipses.

Cross ratios:

• Component-wise cross ratios are obviously preserved.
• Geometric cross-ratios (i.e. for 4 collinear points the cross-ratios of distances between them) are also preserved, as they would be with any conic or linear projection (with the exception of the case when the points are on a line normal to the $x+y+z=1$ plane passing through the origin.)

If you're interested in some other kind of cross-ratio, you have to define how you intend to "multiply" and "divide" the differences of points.

Edit: See corrections.