


\documentclass[xcolor=dvipsnames]{beamer}

%sphere, Impossiball, Masterball for Football, hyperbolic
% 45 minutes? but Interactive
\mode<presentation>
%\usecolortheme{seahorse}
\usepackage{xcolor, times,amsmath,color} 
%\usepackage{xcolor,setspace,amsfonts,amsmath,amssymb,graphicx,verbatim}
\usetheme{Singapore}
 %\usetheme{boxes} 
% \usecolortheme{whale}

\useinnertheme{rounded}

\begin{document}

\begin{frame}
\frametitle{Rotation Matrix - Geometry}

$$\begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta)\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) \end{bmatrix} 
\begin{bmatrix} x \\ y \end{bmatrix} = \pause
\begin{bmatrix} \mbox{cos}(\theta) x -\mbox{sin}(\theta) y \\ \mbox{sin}(\theta) x + \mbox{cos}(\theta)y \end{bmatrix} $$

\pause 

$$\begin{bmatrix} 1 \\ 0 \end{bmatrix} \rightarrow \pause \begin{bmatrix} \mbox{cos}(\theta) \\  \mbox{sin}(\theta) \end{bmatrix} 
$$
\pause
Using the unit circle, we see that  the output is on the $y=\mbox{tan}(\theta)x$ line, ie the same length vector but $\theta$ degrees counterclockwise.

$$\begin{bmatrix} 2 \\ 0 \end{bmatrix} \rightarrow \begin{bmatrix} 2 \mbox{cos}(\theta) \\  2 \mbox{sin}(\theta) \end{bmatrix} 
$$

$$\begin{bmatrix} 0 \\ 1 \end{bmatrix} \rightarrow \pause \begin{bmatrix} -\mbox{sin}(\theta) \\  \mbox{cos}(\theta) \end{bmatrix} 
$$


\end{frame}

\begin{frame}
\frametitle{Rotation Matrix - Algebra}

$$\begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta)\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) \end{bmatrix} 
\begin{bmatrix} x \\ y \end{bmatrix} = 
\begin{bmatrix} \mbox{cos}(\theta) x -\mbox{sin}(\theta) y \\ \mbox{sin}(\theta) x + \mbox{cos}(\theta)y \end{bmatrix} $$

\pause 

$$\begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta)\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) \end{bmatrix} 
\begin{bmatrix} r \mbox{cos}(\phi) \\  r \mbox{sin}(\phi)\end{bmatrix} = 
\begin{bmatrix} r\mbox{cos}(\theta)  \mbox{cos}(\phi)  - r \mbox{sin}(\theta)  \mbox{sin}(\phi) \\ r  \mbox{sin}(\theta) \mbox{cos}(\phi) + r \mbox{cos}(\theta) \mbox{sin}(\phi) \end{bmatrix} $$
\pause 
 $$  \hspace*{2cm} = \begin{bmatrix} r\mbox{cos}(\theta + \phi) \\ r  \mbox{sin}(\theta + \phi) \end{bmatrix} 
$$


\end{frame}
%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Transformations of $\mathbb{R}^2$}
\frametitle{Transformations of $\mathbb{R}^2$}

$$ \mbox{Rotation:} \begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta)\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) \end{bmatrix} 
\mbox{Dilation:} \begin{bmatrix} c & 0 \\ 0 &
c \end{bmatrix} 
 \mbox{Horizontal Shear:} \begin{bmatrix} 1 & k \\ 0 &
1 \end{bmatrix} $$


$$ \mbox{Projections:} \mbox{ y=x line:} \begin{bmatrix} \frac{1}{2} &  \frac{1}{2}\\  \frac{1}{2}  &  \frac{1}{2} \end{bmatrix} 
\mbox{x-axis:} \begin{bmatrix} 1 &  0\\  0 & 0 \end{bmatrix} 
\mbox{y-axis:}\begin{bmatrix} 0 &  0\\  0 & 1 \end{bmatrix} 
$$



$$ \mbox{Reflections:}  \mbox{ y=x line:} \begin{bmatrix} 0 &  1\\  1 &  0\end{bmatrix} 
\mbox{x-axis:}\begin{bmatrix} 1 &  0\\  0 & -1 \end{bmatrix} 
\mbox{y-axis:} \begin{bmatrix} -1 &  0\\  0 & 1 \end{bmatrix} 
$$



\begin{itemize}
\item \begin{center}\textcolor{BrickRed}{What important transformation is missing?}\end{center}
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%


\begin{frame}
\frametitle{Transformations of $\mathbb{R}^2$}

$$ \mbox{Rotation Matrix:} \begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta)\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) \end{bmatrix} $$




$$ \mbox{Dilation Matrix:} \begin{bmatrix} c & 0 \\ 0 &
c \end{bmatrix} 
$$

$$ \mbox{Horizontal Shear:} \begin{bmatrix} 1 & k \\ 0 &
1 \end{bmatrix} 
$$

$$ \mbox{Projection Matrix:} \begin{bmatrix} \mbox{cos}(\theta)^2 & \mbox{cos}(\theta)\mbox{sin}(\theta) \\ \mbox{cos}(\theta)\mbox{sin}(\theta)  &
\mbox{sin}(\theta)^2 \end{bmatrix} 
$$

$$ \mbox{Reflection Matrix:} \begin{bmatrix} \mbox{cos}(\theta)& \mbox{sin}(\theta) \\ \mbox{sin}(\theta)  &
-\mbox{cos}(\theta) \end{bmatrix} 
$$

\begin{itemize}
\item \begin{center}\textcolor{BrickRed}{What important transformation is missing?}\end{center}
\end{itemize}

\end{frame}

\begin{frame}

$$ \mbox{Translation Matrix:} \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix} 
\begin{bmatrix} x \\ y \end{bmatrix} = 
\begin{bmatrix} x + h \\ y +k  \end{bmatrix} 
$$

\pause

$$ \mbox{Translation Matrix:} \begin{bmatrix} ? & ? & ? \\ ? & ? & ? \\ 0 & 0 & 1 \end{bmatrix} 
\begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = 
\begin{bmatrix} x + h \\ y +k \\ 1 \end{bmatrix} 
$$

\pause
$$ \mbox{Rotation Matrix:} \begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta) & 0
\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) & 0
\\ 0& 0&1
\end{bmatrix} $$

$$ \mbox{Rotate a Figure about the point} \begin{bmatrix} 4 \\ 9 \end{bmatrix}: $$
$$ 
  \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 9 \\ 0 & 0 & 1 \end{bmatrix} 
\begin{bmatrix} \mbox{cos}(\theta)&-\mbox{sin}(\theta) & 0
\\ \mbox{sin}(\theta)&\mbox{cos}(\theta) & 0
\\ 0& 0&1    \end{bmatrix} 
  \begin{bmatrix} 1 & 0 & -4 \\ 0 & 1 & -9 \\ 0 & 0 & 1 \end{bmatrix} 
   \begin{bmatrix} x_1 & ... & x_p\\ y_1 &... &  y_p\\ 1 & ... & 1 \end{bmatrix} 
$$




\end{frame}

\end{document}