This document was prepared in \LaTeXe.
- \item[\texttt{DVI}]
- \TeX \, traditionally outputs to the \texttt{DVI} format: a binary format which consists of a
+ \item[DVI]
+ \TeX \, traditionally outputs to the \texttt{DVI} (``DeVice Independent'') format: a binary format which consists of a
simple stack machine with instructions for drawing glyphs and curves\cite{fuchs1982theformat}.
A \texttt{DVI} file is a representation of a document which has been typeset, and \texttt{DVI}
viewers will rasterize this for display or printing, or convert it to another similar format like PostScript
to be rasterized.
- \item[\texttt{HTML}]
+ \item[HTML]
+ The Hypertext Markup Language (HTML)\cite{html2rfc} is the widely used document format which underpins the
+ world wide web. In order for web pages to adapt appropriately to different devices, the HTML format simply
+ defined semantic parts of a document, such as headings, phrases requiring emphasis, references to images or links
+ to other pages, leaving the \emph{layout} up to the browser, which would also rasterize the final document.
+ The HTML format has changed significantly since its introduction, and most of the layout and styling is now controlled
+ by a set of style sheets in the CSS\cite{css2spec} format.
+ \item[PostScript]
+ Much like DVI, PostScript\cite{plrm} is a stack-based format for drawing vector graphics, though unlike DVI (but like \TeX), PostScript is
+ text-based and turing complete. PostScript was traditionally run on a control board in laser printers, rasterizing pages at high resolution
+ to be printed, though PostScript interpreters for desktop systems also exist, and are often used with printers which do not support PostScript natively.\cite{ghostscript}
+
+ PostScript programs typically embody documents which have been typeset, though as a turing-complete language, some layout can be performed by the document.
+
+ \item[PDF]
+ Adobe's Portable Document Format (PDF)\cite{pdfref17} takes the PostScript rendering model, but does not implement a turing-complete language.
+ Later versions of PDF also extend the PostScript rendering model to support translucent regions via Porter-Duff compositing\cite{porter1984compositing}.
+
+ PDF documents represent a particular layout, and must be rasterized before display.
\end{description}
+\subsection{Precision in Document Formats}
-
-Existing document formats, due to being designed to model paper,
-have limited precision (8 decimal digits for PostScript\cite{plrm}, 5 decimal digits for PDF\cite{pdfref17}).
-This matches the limited resolution of printers and ink, but is limited when compared to what aught to be possible
-with ``zoom'' functionality, which is prevent from working beyond a limited scale factor, lest artefacts appear due
+Existing document formats --- typically due to having been designed for documents printed on paper, which of course has
+limited size and resolution --- use numeric types which can only represent a fixed range and precision.
+While this works fine with printed pages, users reading documents on computer screens using programs
+with ``zoom'' functionality are prevented from working beyond a limited scale factor, lest artefacts appear due
to issues with numeric precision.
+\TeX uses a $14.16$ bit fixed point type (implemented as a 32-bit integer type, with one sign bit and one bit used to detect overflow)\cite{beebe2007extending}.
+This can represent values in the range $[-(2^14), 2^14 - 1]$ with 16 binary digits of fractional precision.
+
+The DVI files \TeX \, produces may use ``up to'' 32-bit signed integers\cite{fuchs1982theformat} to specify the document, but there is no requirement that
+implementations support the full 32-bit type. It would be permissible, for example, to have a DVI viewer support only 24-bit signed integers, though many
+files which require greater range may fail to render correctly.
+
+PostScript\cite{plrm} supports two different numeric types: \emph{integers} and \emph{reals}, both of which are specified as strings. The interpreter's representation of numbers
+is not exposed, though the representation of integers can be divined by a program by the use of bitwise operations. The PostScript specification lists some ``typical limits''
+of numeric types, though the exact limits may differ from implementation to implementation. Integers typically must fall in the range $[-2^{31}, 2^{31} - 1]$,
+and reals are listed to have largest and smallest values of $\pm10^{38}$, values closest to $0$ of $\pm10^{-38}$ and approximately $8$ decimal digits of precision,
+derived from the IEEE 754 single-precision floating-point specification.
+
+Similarly, the PDF specification\cite{pdfref17} stores \emph{integers} and \emph{reals} as strings, though in a more restricted format than PostScript.
+The PDF specification gives limits for the internal representation of values. Integer limits have not changed from the PostScript specification, but numbers
+representable with the \emph{real} type have been specified differently: the largest representable values are $\pm 3.403\times 10^{38}$, the smallest non-zero representable values are
+\footnote{The PDF specification mistakenly leaves out the negative in the exponent here.}
+$\pm1.175 \times 10^{-38}$ with approximately $5$ decimal digits of precision \emph{in the fractional part}.
+Adobe's implementation of PDF uses both IEEE 754 single precision floating-point numbers and (for some calculations, and in previous versions) 16.16 bit fixed-point values.
+
+
\section{Rendering}
Computer graphics comes in two forms: bit-mapped (or raster) graphics, which is defined by an array of pixel colours;
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