-Figures \ref{} show the effect of scaling the grid to different view coordinates using single precision. This illustrates the trade off between precision and range; as the top left corner of the view moves further away from the origin, the width for which the grid appears unaltered decreases, or if the width is kept fixed, then there are fewer locations on the grid that can be correctly transformed from document to view space.
+Figure \ref{grid-precision} illustrates the effect of applying the view transformation \eqref{view-transformation} directly to the grid. When the grid is correctly rendered, as in Figure \ref{grid-precision} a) it appears as a black rectangle. Further from the origin, not all pixels in the grid can be represented and individual lines become visible. As the distance from the origin increases, fewer pixel locations can be represented exactly after performing the view transformation.
+
+An error of 1 ulp is increased by a factor of $10^6$ to end up comparable to the size of the display ($0 \to 1$).
+
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=800px]{figures/grid_0_1e-6.png}
+ \includegraphics[width=800px]{figures/{grid_0.5_1e-6}.png}
+ \includegraphics[width=800px]{figures/grid_1_1e-6.png}
+ \includegraphics[width=800px]{figures/grid_2_1e-6.png}
+ \caption{Effect of applying \eqref{view-transformation} to a grid of lines seperated by 1 pixel \\
+ a) Near origin (denormals) b), c), d) Increasing the exponent of $(v_x,v_y)$ by 1}\label{grid-precision}
+\end{figure}
+
+
+\subsection{Names of programs in figures}
+\begin{itemize}
+ \item single - Single precision IEEE-754 with \eqref{view-transformation} applied directly
+ \item double - Double precision IEEE-754 with \eqref{view-transformation} applied directly
+ \item cumul-single - Single precision IEEE-754 with cumulative transforms to B\'{e}ziers
+ \item cumul-double - Double precision IEEE-754 with cumulative transforms to B\'{e}ziers
+ \item path-single - Single precision IEEE-754 with cumulative transforms to Paths
+ \item path-double - Single precision IEEE-754 with cumulative transforms to Paths
+ \item path-rat - GNU MP Rationals with cumulative transforms to Paths
+\end{itemize}
+
+\subsection{Precision for Fixed View}
+