why is inversion shape preserving ?!?!

[cKleinhuis]

i am right now wondering why the inversion is shape preserving, is it that the angles stay the same, because distances surly change... 

[Syntopia]

Who says they are? For instance, a circle inversion might map a circle into a line.

Conformality is a strictly local property - it just says that if two curves crosses each other with some angle between them at the intersection point, the angle will be same on the map produced by a conformal transformation.

[kram1032]

A lot of what we perceive as the "shape" of an object lies in how the angles of this object change. This is a meassure of local shape and also what a conformal map preserves.
Global shapes, e.g. the part of a "shape" that lies in its area or volume, are not preserved.

[cKleinhuis]

i am referin to the mandelbox definition here
https://sites.google.com/site/mandelbox/what-is-a-mandelbox

where he states that by liouvilles therem all such are shape preserving....
http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29

[kram1032]

Yeah, but as said, angles stay the same, so it's shape-preserving.
Since what we typically think of as shape mostly depends on the angles, in mathematical terms we call this property shape preserving.
It really just means that things look similar and locally undistorted.
Globally, there might be distortions - which is what you meant when you said, that lengths are not preserved.

[Syntopia]

i am referin to the mandelbox definition here
https://sites.google.com/site/mandelbox/what-is-a-mandelbox

where he states that by liouvilles therem all such are shape preserving....
http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29

I think Liouville's theorem states what the possible conformal transformations in 3D are. I don't see see any mentioning of shape. Indeed, Wikipedia describes conformal transformations as angle-preserving, and notes that "Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size.". And as kram noted only local (but not global) undistortion is ensured.

I personally don't think it is correct to talk about inversions as a shape preserving transformation, for instance, it is not included on this list:
http://mathcentral.uregina.ca/QQ/database/QQ.09.98/mcandrew1.html
and I couldn't find any examples, but I might be wrong. At least it would be clearer to say it is locally shape preserving,

For a good illustration of angle-preservation, see this list of conformal map projections:
http://www.progonos.com/furuti/MapProj/Dither/ProjConf/projConf.html

They all conserve the angles (look at the right angles between the longitude and lattitude lines), but the shapes are quite different.
 

[Tglad]

Inversion is shape preserving for small shapes.

small is a mathematical word here, meaning the limit as the shape size goes to 0. i.e. infinitesimally small shapes will look the same through a sphere inversion.

This is shorthand for 'conformal or anti-conformal', the transform of a tiny mug will still look like a mug, but it might be a reflection.

I guess I could write 'locally shape preserving' as Syntopia suggested, 'conformal or anti-conformal' is a bit long winded and technical for the first sentence.