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'Euclidean transformation'
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| Title of object: |
Euclidean transformation |
| Canonical Name: |
EuclideanTransformation |
| Type: |
Definition |
| Created on: |
2006-06-12 19:15:26 |
| Modified on: |
2007-06-05 10:30:04 |
| Classification: |
msc:51A15, msc:51A10, msc:15A04 |
| Defines: |
rotation, reflection, glide reflection |
| Synonyms: |
Euclidean transformation=rigid motion
Euclidean transformation=reflexion |
Revision comment (for changes between this and next version):
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Content:
\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, $E$ has the familiar \PMlinkname{matrix representation}{MatrixRepresentation}
\begin{eqnarray}
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix},
\end{eqnarray}
where $\theta\in \mathbb{R}$. Via this map, any vector is rotated
in the counterclockwise direction by an angle of $\theta$ to another
vector.
\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
\PMlinkname{conjugate}{Conjugate4} 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 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}
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