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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?

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What do you mean by the "usual" properties of a linear fractional transformation? –  Yemon Choi Nov 18 '10 at 3:13
    
Bu "usual" I had in mind: -preservation of cross ratio -circles maps to circles -lines map to lines –  user10917 Nov 18 '10 at 3:15
1  
No, this one is just a central projection onto the plane $x+y+z=1,$ and your earlier one was just a cental projection onto the line $x+y=1.$ Nothing whatsoever is being preserved. Actual LFT's are defined in the complex plane, (a z + b) / ( c z + d). You are not finding anything on line because what you are asking makes no sense. –  Will Jagy Nov 18 '10 at 3:23
    
Well, yes, typically LFTs are thought of in the complex plane but this is not necessary. See, for example, this paper tinyurl.com/22tov9s or perhaps even this dcs.ed.ac.uk/home/mhe/plume/node21.html where LFTs are used to develop some computer arithmetic methods. –  user10917 Nov 18 '10 at 3:33
    
A circle in some plane not parallel to your plane is mapped to an ellipse or a line segment (including endpoints). –  Will Jagy Nov 18 '10 at 3:49

1 Answer 1

up vote 1 down vote accepted

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.

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