non-Euclidean geometry NonEuclideanGeometry 2003-08-29 16:43:33 2007-06-11 04:36:52 Definition geometry hyperbolic geometry Bolyai-Lobachevski geometry elliptic geometry spherical geometry semi-Euclidean geometry % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{similar} A \emph{non-Euclidean geometry} is a \PMlinkescapetext{geometry} in which at least one of the axioms from Euclidean geometry fails. Within this entry, only geometries that are considered to be two-dimensional will be considered. The most common non-Euclidean geometries are those in which the parallel postulate fails; \PMlinkname{i.e.}{Ie}, there is not a unique line that does not intersect a given line through a point not on the given line. Note that this is equivalent to saying that the sum of the angles of a triangle is not equal to $\pi$ radians. If there is more than one such parallel line, the \PMlinkescapetext{geometry} is called \emph{hyperbolic} (or \emph{Bolyai-Lobachevski}). In these \PMlinkescapetext{types} of \PMlinkescapetext{geometries}, the sum of the angles of a triangle is strictly in \PMlinkescapetext{between} $0$ and $\pi$ radians. (This sum is not constant as in Euclidean geometry; it depends on the area of the triangle. See the entry regarding defect for more details.) As an example, consider the disc $\{(x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$ in which a point is similar to the Euclidean point and a line is defined to be a chord (excluding its endpoints) of the (\PMlinkname{circular}{Circle}) boundary. This is the Beltrami-Klein model for $\mathbb{H}^2$. It is relatively easy to see that, in this \PMlinkescapetext{geometry}, given a line and a point not on the line, there are infinitely many lines passing through the point that are parallel to the given line. If there is no parallel line, the \PMlinkescapetext{geometry} is called \emph{spherical} (or \emph{elliptic}). In these \PMlinkescapetext{types} of \PMlinkescapetext{geometries}, the sum of the angles of a triangle is strictly in \PMlinkescapetext{between} $\pi$ and $3\pi$ radians. (This sum is not constant as in Euclidean geometry; it depends on the area of the triangle. See the entries regarding \PMlinkname{defect}{Defect} and area of a spherical triangle for more details.) As an example, consider the surface of the \PMlinkname{unit sphere}{Sphere} $\{(x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2=1 \}$ in which a point is similar to the Euclidean point and a line is defined to be a great circle. (Note that, when a sphere serves as a model of spherical geometry, its radius is typically assumed to be 1.) It is relatively easy to see that, in this \PMlinkescapetext{geometry}, given a line and a point not on the line, it is impossible to find a line passing through the point that does not intersect the given line. Note also that, in spherical geometry, two distinct points do not necessarily determine a unique line; however, two distinct points that are not antipodal always determine a unique line. One final example of a non-Euclidean \PMlinkescapetext{geometry} is \emph{semi-Euclidean geometry}, in which the axiom of Archimedes fails.