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In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very familiar with the terminology)And my question is:

Is there a magnitude of graphs that is preserved by the inversion mapping?

I cannot find one; I know there is a magnitude called the ratio which is preserved by projective transformations, is the ratio the thing I want?
Any help is well appreciated, thanks.

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    $\begingroup$ Angles are preserved. $\endgroup$
    – André Nicolas
    Apr 9, 2011 at 4:44
  • $\begingroup$ @user6312, thanks, and how about the ratio, or how does it change the area, maybe I am not asking a good question here, sorry. $\endgroup$
    – awllower
    Apr 9, 2011 at 5:48

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cross-ratio is preserved: if $A,B,C,D$ are points in the plane (not necessarily lying on a line) then $$\frac{|AB||CD|}{|AC||BD|}$$ is preserved.

Also the sum of the angles at $A$ and $C$ in a quadrilateral $ABCD$ is preserved.

These two properties are combined into one: if you see your plane as the complex plane, so that $A,B,C,D$ become complex numbers, then the complex cross-ratio $(A-B)(C-D)/((A-C)(B-D))$ is (unfortunately not preserved, but) complex conjugated by inversion.

(area is not preserved; as noted by user6312 in the comments, angles between intersecting curves are preserved)

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  • $\begingroup$ C'est magnifique! And especially the part of the view point of the complex plane is really astonishing; maybe it seems trivial to you, but I am not familiar with geometry so that it requires more elaborations; and I will look it up in other contexts, and all in all, thanks very much. $\endgroup$
    – awllower
    Apr 11, 2011 at 11:57

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