\paragraph{b)}

First we linearize the input vector. Thus we have the form:

\[
    \begin{pmatrix} x_0 & y_0 & x_1 & y_1 & x_2 & y_2 &  \cdots & x_4 & y_4\end{pmatrix}^T
\]


The corner-cutting subdivision matrix is then a  matrix of form: \[\text{dim} \cdot |P| \times 2 \cdot |P| \]

\[
    \begin{pmatrix}
    0.75 & 0    &   0.25 & 0    &  & &  & \\
    0   &    0.25&  0   &    0.75   &  & &  & \\
    0.25 & 0 &      0.75 & 0    &  & &   & \\
    0 &  0.25& 0 &   0.75 & & & &    \\
    & & 0.75& 0&        0.25& 0&    &     \\
    & & 0   &    0.25   &0  &    0.75   & &  \\
     && 0.25& 0& 0.75&  0& &     \\
     & & 0 &     0.25 &0 &   0.75& &     \\
 & &  & & \ddots & \ddots & \ddots & \ddots      \\
  & &  & & \ddots & \ddots & \ddots & \ddots     \\
   & &  & & \ddots & \ddots & \ddots & \ddots    \\
    & &  & & \ddots & \ddots & \ddots & \ddots  \\
            0.25 &0&&&  & & 0.75& 0\\
        0       & 0.75& &&  & & 0&       0.25 \\
            0.75& 0 & &  &   & & 0.25& 0\\
     0  & 0.75&  &&&    &  0&    0.25\\
     \end{pmatrix}
 \cdot
            \begin{pmatrix} x_0 & y_0 & x_1 & y_1 & x_2 & y_2 &  \cdots & x_4 & y_4\end{pmatrix}^T
\]

The final Vector can be delinearized again in order to have the Points.

\paragraph{c)}
Extraordinary points are points where the order of these points does not change during the regular subdivision.
\end{document}