A straight line is simply a polynomial of $0$th degree. Splines may be rasterised by sampling of $y(x)$ at a number of points $x_i$ and rendering straight lines between $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$ as discussed in Section \ref{Straight Lines}. More direct algorithms for drawing splines based upon Brasenham and Wu's algorithms also exist\cite{citationneeded}.
-There are many different ways to define a spline. One approach is to specify ``knots'' on the spline and solve for the cooefficients to generate a cubic spline ($n = 3$) passing through the points. Alternatively, special polynomials may be defined using ``control'' points which themselves are not part of the curve; these are convenient for graphical based editors. For example, drawing bezier curves with the mouse is the primary method of constructing paths in the Inkscape SVG editor\cite{inkscape}.
-
+There are many different ways to define a spline. One approach is to specify ``knots'' on the spline and solve for the cooefficients to generate a cubic spline ($n = 3$) passing through the points. Alternatively, special polynomials may be defined using ``control'' points which themselves are not part of the curve; these are convenient for graphical based editors. Bezier splines are the most straight forward way to define a curve in the standards considered in Section \ref{Document Representations}
\subsubsection{Bezier Curves}
\input{chapters/Background_Bezier}
\rephrase{2. Here are the ways documents are structured ... we got here eventually}
-\section{Document Representations}
+\section{Document Representations}\label{Document Representations}
The representation of information, particularly for scientific purposes, has changed dramatically over the last few decades. For example, Brassel's 1979 paper referenced earlier has been produced on a mechanical type writer. Although the paper discusses an algorithm for shading on computer displays, the figures illustrating this algorithm have not been generated by a computer, but drawn by Brassel's assistant\cite{brassel1979analgorithm}. In contrast, modern papers such as Barnes et. al's recent paper on embedding 3d images in PDF documents\cite{barnes2013embeddding} can themselves be an interactive proof of concept.
\item Solves security issues, more efficient
\end{itemize}
+\pagebreak
\subsection{Document Object Model}\label{Document Object Model}
\input{chapters/Background_DOM}
\section{Precision in Modern Document Formats}
-We briefly summarise the requirements of standard document formats in regards to the precision of number representations:
+We briefly summarise the requirements of the standards discussed so far in regards to the precision of mathematical operations:
\begin{itemize}
\item {\bf PostScript} predates the IEEE-754 standard and originally specified a floating point representation with ? bits of exponent and ? bits of mantissa. Version ? of the PostScript standard changed to specify IEEE-754 binary32 ``single precision'' floats.
\item {\bf PDF} has also specified IEEE-754 binary32 since version ?. Importantly, the standard states that this is a \emph{maximum} precision; documents created with higher precision would not be viewable in Adobe Reader.