\begin{definition}[Translation]On considère deux points $A$ et $B$ du plan. 

On appelle \MotDefinition{translation}{} \textbf{qui transforme {\boldmath$A$} en {\boldmath $B$}} la transformation qui, à tout point $M$ du plan, associe
l'unique point $M'$ tel que $[ AM' ]$ et $[ BM ]$ ont même milieu. 
\begin{center}
\begin{tikzpicture}[general, scale=0.4]
\draw [quadrillage] (0,0) grid (11,8);
\coordinate (A) at (1,1); 
\coordinate (B) at (3,5); 
\coordinate (M) at (7,2); 
\coordinate (N) at (9,6);
\coordinate (F) at (10.6, 7);
\draw [loosely dashed] (A)--(F);
\draw [loosely  dashed] (B)--(M); 
\draw [->, epais, color=A1] (A)--(B); 
\draw [->, epais, color=A1] (M)--(N); 
\pointC{A}{A}{below left}
\pointC{B}{B}{above left}
\pointC{M}{M}{below right}
\pointC{N}{M'}{above}
\draw (N) arc (32:52:3);
\draw (N) arc (32:12:3);
\draw (3,2.25) node[rotate=122, color=F1]  {{\boldmath $\approx$}};
\draw (7,4.75) node[rotate=122, color=F1]  {{\boldmath $\approx$}};
\draw (4,4.25) node[rotate=53, color=C1]  {{\boldmath $\infty$}};
\draw (6,2.75) node[rotate=53, color=C1]  {{\boldmath $\infty$}};
\draw (A) node{{\boldmath $\setminus$}};
\draw (M) node{{\boldmath $\setminus$}};
\end{tikzpicture}\hfill
\begin{tikzpicture}[general, scale=0.4]
\draw [quadrillage]  (0,0) grid (16,8);
\coordinate (D) at (1,1); 
\coordinate (A) at (2.2,1.48); 
\coordinate (B) at (5.8,2.92); 
\coordinate (M) at (8.5,4); 
\coordinate (N) at (12.1,5.44);
\coordinate (F) at (16, 7);
\draw [loosely dashed] (D)--(F);
\draw [->, epais, color=A1, line cap=butt] (A)--(B); 
\draw [->, epais, color=A1, line cap=butt] (M)--(N); 
\pointC{A}{A}{below left}
\pointC{B}{B}{above left}
\pointC{M}{M}{below right}
\pointC{N}{M'}{above}
\draw (N) arc (21.8:41.8:3);
\draw (N) arc (21.8:1.8:3);
\draw (A) node{{\boldmath $\setminus$}};
\draw (M) node{{\boldmath $\setminus$}};
\end{tikzpicture}
\end{center}
\end{definition}

\begin{vocabulaire}
\begin{itemize}
\item Le point $M'$ est appelé \textcolor{H1}{\textbf{image}} du point $M$.
\item On dit également que $M$ est le \textcolor{H1}{\textbf{translaté}} de $M'$.
\end{itemize}
\end{vocabulaire}

\begin{remarque}
Une \textbf{transformation} sert à \textbf{modéliser} mathématiquement un mouvement.
\begin{itemize}
\item La \textcolor{H1}{\textbf{symétrie centrale}} est la transformation qui modélise le demi-tour. 
\item La \textcolor{H1}{\textbf{translation}} est la transformation qui modélise le glissement rectiligne. Pour la définir, on indique la direction, le sens et la longueur du mouvement. 
\end{itemize}
\end{remarque}
