EuclideanTransformation 2006-06-12 19:15:26 2007-06-19 21:53:19 Definition translation translate rotation rotate reflection reflect reflexion glide reflection angle of rotation \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 \PMlinkescapeword{addition} \PMlinkescapeword{conjugate} \PMlinkescapeword{expressible} \PMlinkescapeword{term} \PMlinkescapeword{types} Let $V$ and $W$ be Euclidean vector spaces. A \emph{Euclidean transformation} is an affine transformation $E:V\to W$, given by $$E(v)=L(v)+w$$ such that $L$ is an \PMlinkname{orthogonal linear transformation}{OrthogonalTransformation}. As an affine transformation, all affine properties, such as incidence and parallelism are preserved by $E$. In addition, since $E(u-v)=L(u-v)$ and $L$ is an \PMlinkescapetext{orthogonal linear transformation}, $E$ preserves lengths of line segments and \PMlinkname{angles between two line segments}{AngleBetweenTwoLines}. Because of this, a Euclidean transformation is also called a \emph{rigid motion}, which is a popular term used in mechanics. \subsubsection*{Types of Euclidean transformations} There are three main types of Euclidean transformations: \begin{enumerate} \item \textbf{translation}. If $L=I$, then $E$ is just a translation. Any Euclidean transformation can be decomposed into a product of an orthogonal transformation $L(v)$, followed by a translation $T(v)=v+w$. \item \textbf{rotation}. If $w=0$, then $E$ is just an orthogonal transformation. If $\operatorname{det}(E)=1$, then $E$ is called a \emph{rotation}. The orientation of any basis (of $V$) is preserved under a rotation. In the case where $V$ is two-dimensional, a rotation is conjugate to a matrix of the form \begin{eqnarray} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, \end{eqnarray} where $\theta\in \mathbb{R}$. Via this particular (unconjugated) map, any vector emanating from the origin is rotated in the counterclockwise direction by an angle of $\theta$ to another vector emanating from the origin. Thus, if $E$ is conjugate to the matrix given above, then $\theta$ is the \emph{angle of rotation} for $E$. \item \textbf{reflection}. If $w=0$ but $\operatorname{det}(E)=-1$ instead, then $E$ is a called \emph{reflection}. Again, in the two-dimensional case, a reflection is \PMlinkescapetext{conjugate} to a matrix of the form \begin{eqnarray} \begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{pmatrix}, \end{eqnarray} where $\theta\in \mathbb{R}$. Any vector is reflected by this particular (unconjugated) map to another by a ``mirror'', a line of the form $y=x\tan(\frac{\theta}{2})$. \end{enumerate} \textbf{Remarks}. \begin{itemize} \item In general, an orthogonal transformation can be represented by a matrix of the form $$ \begin{pmatrix} A_1 & O & \cdots & O \\ O & A_2 & \cdots & O \\ \vdots & \vdots & \ddots & \vdots \\ O & O & \cdots & A_n \end{pmatrix}, $$ where each $A_i$ is either $\pm 1$ or a rotation matrix (1) (or reflection matrix (2)) given above. When its determinant is -1 (a reflection), it has an invariant subspace of $V$ of codimension 1. One can think of this hyperplane as the mirror. \item Another common rigid motion is the \emph{glide reflection}. It is a Euclidean transformation that is expressible as a product of a reflection, followed by a translation. \end{itemize}