\chapter{Results and Discussion}
+{\bf Note: Need to be more consistent, I often refer to B\'{e}ziers and Objects interchangably (since the original design was based around an Object and B\'{e}zier was just one possible Object, but we have moved on to pretty much only caring about B\'{e}ziers now)}
\section{Qualitative Rendering Accuracy}
Our ultimate goal is to be able to insert detail at an arbitrary point in the document. Therefore, we are interested in how the same test SVG would appear when scaled to the view coordinates, as the view coordinates are varied.
-applying Transformation \eqref{view-transformation}. We will use single precision floats for coordinates unless otherwise stated.
-Figure \ref{qualitative-rendering-fox} shows the rendering of a vector image\footnote{Unfortunately, since a rendered vector image is a raster image and this figure must be scaled to fit the PDF, the figure as seen here is not a pixel perfect representation of the actual rendering. Most notably, antialiasing effects will be apparent}. Transformation \eqref{view-transformation} is applied to object coordinates with default IEEE-754 rounding behaviour (to nearest).
+\subsection{Applying the view transformation directly}
+
+Figure \ref{qualitative-rendering-fox} shows the rendering of a vector image\footnote{Unfortunately, since a rendered vector image is a raster image and this figure must be scaled to fit the PDF, the figure as seen here is not a pixel perfect representation of the actual rendering. Most notably, antialiasing effects will be apparent}. Transformation \eqref{view-transformation} is applied to object coordinates with default IEEE-754 rounding behaviour (to nearest). The loss of precision in the second figure is obvious. This is because division by $10^{-6}$ increases the rounding error in $x - v_x$, by $10^{6}$, so the total error is of the order $10^6$ ulp which is of the order $0.25$
+
+{\bf TODO: Calculate that properly, shouldn't be hard}
\caption{The vector image from Figure \ref{vector-vs-raster} under two different scales}\label{qualitative-rendering-fox}
\end{figure}
-Rather than applying \eqref{view-transformation} to object coordinates specified relative to the document, we can store the bounds of objects relative to the view and modify these bounds according to transformations \eqref{} and \eqref{} as the view is changed. This approach significantly improves the range over which an image can be rendered correctly, and it is convenient for interactively created documents as no transformation must be applied to add new detail.
+\subsection{Applying cumulative transformations to all B\'{e}ziers}
-However, repeated transformations on the view will cause an accumulated error on the coordinates of object bounds. This is most noticable when zooming out and then back into the document; the object bounds coordinates will gradually underflow and eventually round to zero. An example of this effect is shown in Figure \ref{qualitative-rendering-fox-cumulative} b)
+Rather than applying \eqref{view-transformation} to object coordinates specified relative to the document, we can store the bounds of objects relative to the view and modify these bounds according to transformations \eqref{} and \eqref{} as the view is changed. This is convenient for an interactive document, as detail is typically added by inserting objects into the document within the view rectangle. As a result this approach makes the rendering of detail added to the document independent of the view coordinates --- until the view is moved.
+
+Repeated transformations on the view will cause an accumulated error on the coordinates of object bounds. This is most noticable when zooming \emph{out} and then back into the document; the object coordinates will gradually underflow and eventually round to zero. An example of this effect is shown in Figure \ref{qualitative-rendering-fox-cumulative} b)
%label start
%setbounds 0.5 0.5 1e-6 1e-6
%loadsvg svg-tests/fox-vector.svg
\caption{The effect of applying cumulative transformations to all B\'{e}ziers}\label{qualitative-rendering-fox-cumulative}
\end{figure}
-In Figure \ref{qualitative-rendering-fox}, transformations are applied to the bounds of each B\'{e}zier. Figure \ref{qualitative-rendering-fox-cumulative-relative} a) shows the effect of introducing an intermediate coordinate system expressing B\'{e}zier coordinates relative to the path which contains them. In this case, the rendering of a single path is accurate, but the overall positions of the paths drift.
+\subsection{Applying cumulative transformations to Paths}
+
+In Figure \ref{qualitative-rendering-fox}, transformations are applied to the bounds of each B\'{e}zier. Figure \ref{qualitative-rendering-fox-cumulative-relative} a) shows the effect of introducing an intermediate coordinate system expressing B\'{e}zier coordinates relative to the path which contains them. In this case, the rendering of a single path is accurate, but the overall positions of the paths drift as the view is moved.
-We can correct this drift whilst maintaining performance by using an arbitrary precision number representation to express the coordinates of the paths - but maintaining the floating point coordinates for B\'{e}zier curves relative to their path. As we will discuss in Section \ref{}, this offers an acceptable trade off between rendering accuracy and performance.
+We can correct this drift whilst maintaining performance by using an arbitrary or high precision number representation to express the coordinates of the paths - but maintaining the floating point coordinates for B\'{e}zier curves relative to their path. As we will discuss in Section \ref{}, this offers an acceptable trade off between rendering accuracy and performance.
\begin{figure}[H]
\centering
A useful test SVG is a simple grid of horizontal and vertical lines seperated by 1 pixel. When this SVG is correctly scaled to a view, all that should be visible is a coloured rectangle filling the screen. Increasing the magnification will reveal the grid of lines indicating how the original size of a pixel is scaled.
-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}
+
-Figure \ref{} shows a plot of the number of lines visible in the grid as a function of distance from the origin for IEEE-754 floats and several different precision settings for the MPFR library.
+By counting the number of distinctly representable lines within a particular view, we can show the degradation of precision quantitatively. The test grid is added to each view rectangle.
-Figure \ref{} shows the obvious
+
+Figure \ref{loss_of_precision_grid_0.5.pdf} shows how precision degrades with $(v_x, v_y) = (0.5,0.5)$.
+A constant line at $1401$ grid locations indicates no loss of precision.
+
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{{figures/loss_of_precision_grid_0.5}.pdf}
+ \caption{Loss of precision of the grid}
+ \label{loss_of_precision_grid_0.5.pdf}
+\end{figure}
+
+\subsection{Accumulated error after changing the View}
+
+Figure \ref{cumulative_error_grid.pdf} shows the total error in the coordinates of each line in the grid after the view is scaled (zooming \emph{out}) by repeated transformations. A constant line at $0$ indicates no accumulated error.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/cumulative_error_grid.pdf}
+ \caption{Error in the coordinates of the grid}
+ \label{cumulative_error_grid.pdf}
+\end{figure}
+
+By considering Figure \ref{loss_of_precision_grid_0.5.pdf} and \ref{cumulative_error_grid.pdf}, \verb/path-rat/ is the winner.
\section{Performance Measurements whilst Rendering}
As discussed above, we succeeded in preserving rendering accuracy as defined above for an arbitrary view.
However this comes at a performance cost, as the size of the number representation must grow accordingly.
+{\bf TODO: Insert performance measurements here}
-
+{\bf TODO: Also, would be nice to show a graph (log scale) where something goes past $10^{\pm320}$ (absolute limit for doubles, previous figures are all within range of representable floats}
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