defect Defect 2006-07-19 17:18:59 2007-06-26 02:01:00 Definition % 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 Consider a triangle $\triangle ABC$ in either \PMlinkname{hyperbolic or spherical geometry}{NonEuclideanGeometry} in which its angle sum in radians is $\Sigma$. In hyperbolic geometry, the \emph{defect} of $\triangle ABC$ is $\delta(\triangle ABC)=\pi-\Sigma$. In spherical geometry, the \emph{defect} of $\triangle ABC$ is $\delta(\triangle ABC)=\Sigma-\pi$. Note that, in both hyperbolic and spherical geometry, the area of a \PMlinkescapetext{triangle} is equal to its defect.