Conformal space
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) |
Conformal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_space&oldid=32032