projectivity Projectivity 2006-06-09 12:49:43 2006-06-17 15:01:26 Definition collineation projective transformation projective property \usepackage{amssymb,amscd} \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} % define commands here Let $PG(V)$ and $PG(W)$ be projective geometries, with $V,W$ vector spaces over a field $K$. A function $p$ from $PG(V)$ to $PG(W)$ is called a \emph{projective transformation}, or simply a \emph{projectivity} if \begin{enumerate} \item $p$ is a bijection, and \item $p$ is order preserving. \end{enumerate} A \emph{projective property} is any geometric property, such as incidence, linearity, etc... that is preserved under a projectivity. From the definition, we see that a projectivity $p$ carries 0 to 0, $V$ to $W$. Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short, $p$ preserves linearity. Because $p$ is a bijection, $p$ also preserves dimensions, that is $\operatorname{dim}(S)=\operatorname{dim}(p(S))$, for any subspace $S$ of $V$. Other properties preserved by $p$ are incidence: if $S\cap T\neq \varnothing$, then $p(S)\cap p(T)\neq \varnothing$; and \PMlinkname{cross ratios}{CrossRatio}. Every bijective semilinear transformation defines a projectiviity. To see this, let $f:V\to W$ be a semilinear transformation. If $S$ is a subspace of $V$, then $f(S)$ is a subspace of $W$, as $x,y\in f(S)$, then $x+y=f(a)+f(b)=f(a+b)\in f(S)$, and $\alpha x={\beta}^{\theta}x={\beta}^{\theta}f(a)=f(\beta a)\in f(S)$, where $\theta$ is an automorphism of the common underlying field $K$. Also, if $S$ is a subspace of a subspace $T$ of $V$, then $f(S)$ is a subspace of $f(T)$. Now if we define $f^*:PG(V)\to PG(W)$ by $f^*(S)=f(S)$, it is easy to see that $f^*$ is a projectivity. Conversely, if $V$ and $W$ are of finite dimension greater than $2$, then a projectivity $p:PG(V)\to PG(W)$ induces a semilinear transformation $\hat{p}:V\to W$. This highly non-trivial fact is the (first) fundamental theorem of projective geometry. If the semilinear transformation induced by the projectivity $p$ is in fact a linear transformation, then $p$ is a {\it collineation}: three distinct collinear points are mapped to three distinct collinear points. \textbf{Remark}. The definition given in this entry is a generalization of the definition typically given for a projective transformation. In the more restictive definition, a projectivity $p$ is defined merely as a bijection between two projective spaces that is induced by a linear isomorphism. More precisely, if $P(V)$ and $P(W)$ are projective spaces induced by the vector spaces $V$ and $W$, if $L:V\to W$ is a bijective linear transformation, then $p=P(L):P(V)\to P(W)$ defined by $$P(L)[v]=[Lv]$$ is the corresponding projective transformation. $[v]$ is the homogeneous coordinate representation of $v$. In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases, $p$ is expressible in terms of an invertible matrix ($Lv=Av$ where $A$ is an invertible matrix).