Is it possible to point-invert a Euclidean space and not produce a hyperbolic one?
2026-02-22 22:54:19.1771800859
Inversive geometry: is it possible to point-invert a Euclidean space and not produce a hyperbolic one?
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Inversion is a transformation in Euclidean geometry. If a circle $C$ is given then all the points inside the circle will be transformed to points outside of $C$ and vice versa. The points on the perimeter of $C$ will stay.
This way you don't get hyperbolic geometry.
However if you want to construct arcs of circles within $C$ that are perpendicular to $C$ then you will use inversion as a construction tool.
Now, you have arcs within $C$ but this is still not hyperbolic geometry. Further to all said above you will have to appropriately define congruence between segments on these arcs within $C$.
Then you get to the Poincaré model of hyperbolic geomtry.