matrix_rotation_notes_20131214.tex

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\fancyhead[RO]{Branner: Matrix Transformation Notes}
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\title{Matrix Transformation Notes}
\author{David Prager Branner}
\date{14 December, 2013}


\begin{document}
\maketitle

\section{Rotation}

Make a model of how the various kinds of points move when a matrix is rotated:

% side by side figures
% code after http://tex.stackexchange.com/a/1645/3935
\setcapwidth[c]{.2\textwidth}			% positions whole caption
\begin{figure}[htbp]
 \centering%
 \begin{minipage}[b]{.3\linewidth}
  \centering%
   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
    a & f & b \\ 
    (0,0) & (1,0) & (2,0) \\ \hline
    & e & \\ 
    \dots & (1,1) & \dots \\ \hline
    d & & c \\ 
    (0,2) & \dots & (2,2) \\ \hline
   \end{tabular}
  \caption{Initial state}
 \end{minipage}%
% \hfill%
 \begin{minipage}[b]{.05\linewidth}
  \centering%
   \(\longrightarrow\)
   \quad \\
   \quad \\
   \quad \\
   \quad \\
   \quad \\
 \end{minipage}%
% \hfill%
 \begin{minipage}[b]{.3\linewidth}
  \centering%
   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
    d &  & a \\ 
    (0,0) & \dots & (2,0) \\ \hline
    & e & f \\ 
    \dots & (1,1) & (2,1) \\ \hline
    c & \dots & b \\ 
    (0,2) & & (2,2) \\ \hline
   \end{tabular}
  \caption{After 90° rotation clockwise}
 \end{minipage}
\end{figure}
\setcapwidth[c]{.7\textwidth}			% positions whole caption

It may be useful to tabulate these single-point movements, to make your function easier to check:

\begin{table}[htdp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|}\hline
   type & label & location & location \\ \hline
   corner & a & (0,0) & (2,0) \\ \hline
   corner & b & (2,0) & (2,2) \\ \hline
   corner & c & (2,2) & (0,2) \\ \hline
   corner & d & (0,2) & (0,0) \\ \hline
   center & e & (1,1) & (1,1) \\ \hline
   non-corner, non-center & f & (1,0) & (2,1) \\ \hline
  \end{tabular}
 \end{center}
\end{table}%

This rotation can be effected by applying the following code to every point in the matrix:

\begin{lstlisting}
def rotate_point(x, y, side_length):
    return ((side_length - 1 - y), x)
\end{lstlisting}

In the example above, \texttt{side\_length = 3}. Check the results of the function against the various edge cases.

%\newpage
%\subsection*{Counterclockwise rotation}
%\setcapwidth[c]{.2\textwidth}			% positions whole caption
%\begin{figure}[htbp]
% \begin{minipage}[b]{.3\linewidth}
% \centering%
%   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
%    a & f & b \\ 
%    (0,0) & (1,0) & (2,0) \\ \hline
%    & e & \\ 
%    \dots & (1,1) & \dots \\ \hline
%    d & & c \\ 
%    (0,2) & \dots & (2,2) \\ \hline
%   \end{tabular}
%  \caption{Initial state}
% \end{minipage}%
%% \hfill%
% \begin{minipage}[b]{.05\linewidth}
%  \centering%
%   \(\longrightarrow\)
%   \quad \\
%   \quad \\
%   \quad \\
%   \quad \\
%   \quad \\
% \end{minipage}%
%% \hfill%
% \begin{minipage}[b]{.3\linewidth}
%  \centering%
%   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
%    b & & c \\ 
%    (0,0) & \dots & (2,0) \\ \hline
%    f & e & \\ 
%    (0,1) & (1,1) & \dots \\ \hline
%    a & & d \\ 
%    (0,2) & \dots & (2,2) \\ \hline
%   \end{tabular}
%  \caption{After 90° rotation counterclockwise}
% \end{minipage}
%\end{figure}
%\setcapwidth[c]{.7\textwidth}			% positions whole caption
%
%This rotation can be effected by applying the following code to every point in the matrix:

Similarly, the following will bring about counterclockwise rotation:

\begin{lstlisting}
def rotate_point(x, y, side_length):
    return (y, (side_length - 1 - x))
\end{lstlisting}

%\begin{table}[htdp]
% \begin{center}
%  \begin{tabular}{|c|c|c|c|}\hline
%   type & label & location & location \\ \hline
%   corner & a & (0,0) & (0,2) \\ \hline
%   corner & b & (2,0) & (0,0) \\ \hline
%   corner & c & (2,2) & (2,0) \\ \hline
%   corner & d & (0,2) & (2,2) \\ \hline
%   center & e & (1,1) & (1,1) \\ \hline
%   non-corner, non-center & f & (1,0) & (0,1) \\ \hline
%  \end{tabular}
% \end{center}
%\end{table}%

\section{Flipping}

Flipping is a different kind of transformation from rotation, because it turns the ``face'' of the matrix away from us. We can flip along a row (``vertically''), a column, or a diagonal. Let's use the diagonal running from lower left to upper right. As before, make a model:

% side by side figures
% code after http://tex.stackexchange.com/a/1645/3935
\setcapwidth[c]{.2\textwidth}			% positions whole caption
\begin{figure}[htbp]
 \centering%
 \begin{minipage}[b]{.3\linewidth}
 \centering%
   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
    a & f & b \\ 
    (0,0) & (1,0) & (2,0) \\ \hline
    & e & \\ 
    \dots & (1,1) & \dots \\ \hline
    d & & c \\ 
    (0,2) & \dots & (2,2) \\ \hline
   \end{tabular}
  \caption{Initial state}
 \end{minipage}%
% \hfill%
 \begin{minipage}[b]{.05\linewidth}
  \centering%
   \(\longrightarrow\)
   \quad \\
   \quad \\
   \quad \\
   \quad \\
   \quad \\
 \end{minipage}%
% \hfill%
 \begin{minipage}[b]{.3\linewidth}
  \centering%
   \begin{tabular}{|>{\centering}p{.35in}|>{\centering}p{.35in}|>{\centering}p{.35in}|}\hline
    c &  & b \\ 
    (0,0) & \dots & (2,0) \\ \hline
     & e & f \\ 
    \dots & (1,1) & (2,1) \\ \hline
    d & & a \\ 
    (0,2) & \dots & (2,2) \\ \hline
   \end{tabular}
  \caption{After 90° rotation clockwise}
 \end{minipage}
\end{figure}
\setcapwidth[c]{.7\textwidth}			% positions whole caption

Tabulate for reference:

\begin{table}[htdp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|}\hline
   type & label & location & location \\ \hline
   corner & a & (0,0) & (2,2) \\ \hline
   corner & b & (2,0) & (2,0) \\ \hline
   corner & c & (2,2) & (0,0) \\ \hline
   corner & d & (0,2) & (0,2) \\ \hline
   center & e & (1,1) & (1,1) \\ \hline
   non-corner, non-center & f & (1,0) & (2,1) \\ \hline
  \end{tabular}
 \end{center}
\end{table}%

Note that the points forming the diagonal on which we are flipping do not change position. All the other points move to a place symmetrically opposite their place of origin across the diagonal. 

\begin{lstlisting}
def flip_on_diag_lowerleft_to_upperright(x, y, side_length):
    # points change place only if not on this diagonal
    if x + y !== side_length:
        return side_length - 1 - y, side_length - 1 - x
    return x, y
\end{lstlisting}


\end{document}