Conformal space

$M_n$

The Euclidean space $E_n$ extended by an ideal point (at infinity). It is considered in conformal geometry. The fundamental group corresponding to this space consists of point transformations taking spheres (circles in $M_2$) into spheres. By means of stereographic projection, the conformal space $M_n$ is mapped onto the absolute $K_n$ of the space $P_{n+1}$ with a hyperbolic metric. The fundamental group of conformal geometry is isomorphic to the group of hyperbolic motions of this space $P_{n+1}$.

The presence of the ideal point ensures that the stereographic projection is one-to-one. Under transformations of the conformal group, the ideal point can be taken to an ordinary point. Therefore, in a conformal space a sphere is indistinguishable from a plane: A plane is a sphere passing through the ideal point.

Comments

Conformal geometry is also called Möbius geometry, and the absolute of $P_{n+1}$ is also called the absolute quadric of $P_{n+1}$.

More about the geometry of $M_2$ can be found in [a1].

References

[a1]	H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)