Difference between revisions of "Conformal space"
From Encyclopedia of Mathematics
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| − | The Euclidean space | + | The Euclidean space $E_n$ extended by an ideal point (at infinity). It is considered in [[Conformal geometry|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|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. | 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. | ||
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| − | Conformal geometry is also called Möbius geometry, and the absolute of | + | 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 | + | More about the geometry of $M_2$ can be found in [[#References|[a1]]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)</TD></TR></table> | ||
Latest revision as of 14:38, 1 May 2014
$M_n$
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) |
How to Cite This Entry:
Conformal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_space&oldid=17973
Conformal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_space&oldid=17973
This article was adapted from an original article by G.V. Bushmanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article