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}
+\subsection{Font Rendering}
+
+Donald Knuth's 1986 textbook ``Metafont'' blargh
+
+
+
\subsection{Shading}
Algorithms for shading on vector displays involved drawing equally spaced lines in the region with endpoints defined by the boundaries of the region\cite{brassel1979analgorithm}. Apart from being unrealistic, these techniques required a computationally expensive sorting of vertices\cite{lane1983analgorithm}.
On raster displays, shading is typically based upon Lane's algorithm of 1983\cite{lane1983analgorithm}. Lane's algorithm relies on the ability to ``subtract'' fill from a region. This algorithm is now implemented in the GPU \rephrase{stencil buffer-y and... stuff} \cite{kilgard2012gpu}
-\subsection{Compositing}
+\subsection{Compositing and the Painter's Model}\label{Compositing and the Painter's Model}
So far we have discussed techniques for rendering vector graphics primitives in isolation, with no regard to the overall structure of a document which may contain many thousands of primitives. A straight forward approach would be to render all elements sequentially to the display, with the most recently drawn pixels overwriting lower elements. Such an approach is particularly inconvenient for anti-aliased images where colours must appear to smoothly blur between the edge of a primitive and any drawn underneath it.
-Most raster displays are based on an additive red-green-blue colour representation which matches the human eye's response to light\cite{citationneeded}. In 1984, Porter and Duff introduced a fourth colour channel to be used when combining rasterised images called the ``alpha'' channel, analogous to the transparency of a pixel\cite{porter1984compositing}. Elements can be rendered seperately, with the four colour channels of successively drawn elements being combined according to one of several possible operations described by Porter and Duff.
+Colour raster displays are based on an additive red-green-blue $(r,g,b)$ colour representation which matches the human eye's response to light\cite{computergraphics2}. In 1984, Porter and Duff introduced a fourth colour channel for rasterised images called the ``alpha'' channel, analogous to the transparency of a pixel\cite{porter1984compositing}. In compositing models, elements can be rendered seperately, with the four colour channels of successively drawn elements being combined according to one of several possible operations.
+
+In the ``painter's model'' as described by the SVG standard, Porter and Duff's ``over'' operation is used when rendering one primitive over another\cite{svg2011-1.1}.
+Given an existing pixel $P_1$ with colour values $(r_1, g_1, b_1, a_1)$ and a pixel $P_2$ with colours $(r_2, g_2, b_2, a_2)$ to be painted over $P_1$, the resultant pixel $P_T$ has colours given by:
+\begin{align}
+ a_T &= 1 - (1-a_1)(1-a_2) \\
+ r_T &= (1 - a_2)r_1 + r_2 \quad \text{(similar for $g_T$ and $b_T$)}
+\end{align}
+It should be apparent that alpha values of $1$ correspond to an opaque pixel; that is, when $a_2 = 1$ the resultant pixel $P_T$ is the same as $P_2$.
+When the final pixel is actually drawn on an rgb display, the $(r, g, b)$ components are $(r_T/a_T, g_T/a_T, b_T/a_T)$.
-In the ``painter's model'' described by the SVG standard, the ``over'' operation is used when rendering one primitive over another; the red-green-blue components of overlapping pixels are added but the alpha component is set to that of the uppermost pixel\cite{svg2011-1.1}. The PostScript and PDF standards also use the ``painter's model''. The painter's model is demonstrated in Figure \ref{SVG} --- originally an SVG image but converted to a PDF for inclusion in this report\footnote{PDF and SVG formats may be converted but neither standard allows for importing the other directly}.
+The PostScript and PDF standards, as well as the OpenGL API also use a painter's model for compositing. However, PostScript does not include an alpha channel, so $P_T = P_2$ always\cite{plrm}. Figure \ref{SVG} illustrates the painter's model for partially transparent shapes as they would appear in both the SVG and PDF models.
\subsection{Rasterisation on the CPU and GPU}
\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.
-\rephrase{Say some stuff about Knuth's Metafont and \TeX here}
+In this section we will consider various approaches and motivations to specifying the structure and appearance of a document, including: early interpreted formats (PostScript, \TeX, DVI), the Document Object Model popular in standards for web based documents (HTML, SVG), and Adobe's ubiquitous Portable Document Format (PDF). Some of these formats were discussed in a recent paper ``Pixels Or Perish'' by Hayes\cite{hayes2012pixelsor} who argues for greater interactivity in the PDF standard.
-Hayes' 2012 article ``Pixels or Perish'' discusses the recent history and current state of the art in documents for scientific publications\cite{hayes2012pixels}.
+\subsection{Interpreted Document Formats}
+\input{chapters/Background_Interpreted}
-\subsection{Interpreted Model}
\begin{itemize}
\item This model treats a document as the source code program which produces graphics
\item Problems with security --- Turing complete, can be exploited easily
\end{itemize}
-\subsection{Crippled Interpreted Model}
-
-\rephrase{I'm pretty sure I made that one up}
-
-\begin{itemize}
- \item PDF is PostScript but without the Turing Completeness
- \item Solves security issues, more efficient
-\end{itemize}
-
+\pagebreak
\subsection{Document Object Model}\label{Document Object Model}
\input{chapters/Background_DOM}
+\subsection{The Portable Document Format}
+
+
\subsection{Scientific Computation Packages}
The document and the code that produces it are one and the same.
\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.
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