\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{hyperref}
+\usepackage{graphicx}
+\usepackage{amsmath}
+\usepackage{amssymb}
%opening
\title{Literature Review}
And thus the document was born.
-Traditionally, documents have been static: just marks on paper, but with the advent of computers many more possibilities open up.
-Most existing document formats --- such as the venerable PostScript and PDF --- are, however, designed to imitate
-existing paper documents, largely to allow for easy printing. In order to truly take advantage of the possibilities operating in the digital
-domain opens up to us, we must look to new formats.
+Traditionally, documents have been static: just marks on rock, parchment or paper, but with the advent of computers many more possibilities open up.
-Formats such as \texttt{HTML} allow for a greater scope of interactivity and for a more data-driven model, allowing
-the content of the document to be explored in ways that perhaps the author had not anticipated.\cite{hayes2012pixels}
-However, these data-driven formats typically do not support fixed layouts, and the display differs from renderer to
-renderer.
-
-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
-to issues with numeric precision.
\section{Rendering}
-Computer graphics comes in two forms: bit-mapped (or raster) graphics, which is defined by an array of pixel colours,
+Computer graphics comes in two forms: bit-mapped (or raster) graphics, which is defined by an array of pixel colours;
and \emph{vector} graphics, defined by mathematical descriptions of objects. Bit-mapped graphics are well suited to photographs
-and are match how cameras, printers and monitors work. However, bitmap devices do not handle zooming beyond their
-``native'' resolution --- the resolution where one document pixel maps to one display pixel ---, exhibiting an artefact
-called pixelation where the pixel structure becomes evident. Attempts to use interpolation to hide this effect are
-never entirely successful, and sharp edges, such as those found in text and diagrams, are particularly effected.
+and match how cameras, printers and monitors work.
+
+
+\begin{figure}[h]
+ \centering \includegraphics[width=0.8\linewidth]{figures/vectorraster_example}
+ \caption{A circle as a vector image and a $32 \times 32$ pixel raster image}
+\end{figure}
+
+
+However, bitmap devices do not handle zooming beyond their ``native'' resolution (the resolution where one document pixel maps
+to one display pixel), exhibiting an artefact called pixelation where the pixel structure becomes evident. Attempts to use
+interpolation to hide this effect are never entirely successful, and sharp edges, such as those found in text and diagrams, are particularly affected.
-Vector graphics lack many of these problems: the representation is independent of the output resolution, and rather
+Vector graphics avoid many of these problems: the representation is independent of the output resolution, and rather
an abstract description of what it is being rendered, typically as a combination of simple geometric shapes like lines,
-arcs and ``B\'ezier curves''.
+arcs and glyphs.
+
As existing displays (and printers) are bit-mapped devices, vector documents must be \emph{rasterized} into a bitmap at
a given resolution. This bitmap is then displayed or printed. The resulting bitmap is then an approximation of the vector image
at that resolution.
+% Project specific line
This project will be based around vector graphics, as these properties make it more suited to experimenting with zoom
quality.
-
-The rasterization process typically operates on an individual ``object'' or ``shape'' at a time: there are special algorithms
-for rendering lines\cite{bresenham1965algorithm}, triangles\cite{giesen2013triangle}, polygons\cite{pineda1988parallel} and B\'ezier
-Curves\cite{goldman_thefractal}. Typically, these are rasterized independently and composited in the bitmap domain using Porter-Duff
-compositing\cite{porter1984compositing} into a single image. This allows complex images to be formed from many simple pieces, as well
-as allowing for layered translucent objects, which would otherwise require the solution of some very complex constructive geometry problems.
-
-\subsection{Compositing Digital Images\cite{porter1984compositing}}
-
-
-
-Perter and Duff's classic paper "Compositing Digital Images" lays the
-foundation for digital compositing today. By providing an "alpha channel,"
-images of arbitrary shapes — and images with soft edges or sub-pixel coverage
-information — can be overlayed digitally, allowing separate objects to be
-rasterized separately without a loss in quality.
-
-Pixels in digital images are usually represented as 3-tuples containing
-(red component, green component, blue component). Nominally these values are in
-the [0-1] range. In the Porter-Duff paper, pixels are stored as $(R,G,B,\alpha)$
-4-tuples, where alpha is the fractional coverage of each pixel. If the image
-only covers half of a given pixel, for example, its alpha value would be 0.5.
-
-To improve compositing performance, albeit at a possible loss of precision in
-some implementations, the red, green and blue channels are premultiplied by the
-alpha channel. This also simplifies the resulting arithmetic by having the
-colour channels and alpha channels use the same compositing equations.
-
-Several binary compositing operations are defined:
+\subsection{Rasterizing Vector Graphics}
+
+Before an vector document can be rasterized, the co-ordinates of any shapes must
+be transformed into \emph{screen space} or \emph{viewport space}\cite{blinn1992trip}.
+On a typical display, many of these screen-space coordinates require very little precision or range.
+However, the co-ordinate transform must take care to ensure that precision is preserved during this transform.
+
+After this transformation, the image is decomposed into its separate shapes, which are rasterized
+and then composited together.
+Most graphics formats support Porter-Duff compositing\cite{porter1984compositing}.
+Porter-Duff compositing gives each element (typically a pixel) a ``coverage'' value,
+denoted $\alpha$ which represents the contribution of that element to the final scene.
+Completely transparent elements would have an $\alpha$ value of $0$, and completely opaque
+elements have an $\alpha$ of $1$. This permits arbitrary shapes to be layered over one another
+in the raster domain, while retaining soft-edges.
+
+The rasterization process may then proceed on one object (or shape) at a time. There are special algorithms for rasterizing
+different shapes.
+
+\begin{description}
+ \item[Line Segment]
+ Straight lines between two points are easily rasterized using Bresenham's algorithm\cite{bresenham1965algorithm}.
+ Bresenham's algorithm draws a pixel for every point along the \emph{long} axis of the line, moving along the short
+ axis when the error exceeds $\frac{1}{2}$ a pixel.
+
+ Bresenham's algorithm only operates on lines whose endpoints lie on integer pixel coordinates. Due to this, line ``clipping''
+ may be performed to find endpoints of the line segment such that the entire line will be on-screen. However, if line clipping is
+ performed na\"ively without also setting the error accumulator correctly, the line's slope will be altered slightly, becoming dependent
+ on the viewport.
+
+ \item[B\'ezier Curve]
+ A B\'ezier curve is a smooth (i.e.\ infinitely differentiable) curve between two points, represented by a Bernstein polynomial.
+ The coefficients of this Bernstein polynomial are known as the ``control points.''
+
+ B\'ezier curves are typically rasterized using De Casteljau's algorithm\cite{foley1996computer}
+ Line Segments are a first-order B\'ezier curve.
+
+ \item[B\'ezier Spline]
+ A spline of order $n$ is a $C^{n-1}$ smooth continuous piecewise function composed of polynomials of degree $\leq n$.
+ In a B\'ezier spline, these polynomials are Bernstein polynomials, hence the spline is a curve made by joining B\'ezier curves
+ end-to-end (in a manner which preserves some level of smoothness).
+
+ Many vector graphics formats call B\'ezier splines of a given order (typically quadratic or cubic) ``paths'' and treat them as the
+ fundamental type from which shapes are formed.
+\end{description}
+
+
+%There are special algorithms
+%for rendering lines\cite{bresenham1965algorithm}, triangles\cite{giesen2013triangle}, polygons\cite{pineda1988parallel} and B\'ezier
+%Curves\cite{goldman_thefractal}.
+
+\subsection{GPU Rendering}
+While traditionally, rasterization was done entirely in software, modern computers and mobile devices have hardware support for rasterizing
+lines and triangles designed for use rendering 3D scenes. This hardware is usually programmed with an
+API like \texttt{OpenGL}\cite{openglspec}.
+
+More complex shapes like B\'ezier curves can be rendered by combining the use of bitmapped textures (possibly using signed-distance
+fields\cite{leymarie1992fast}\cite{frisken2000adaptively}\cite{green2007improved}) strtched over a triangle mesh
+approximating the curve's shape\cite{loop2005resolution}\cite{loop2007rendering}.
+
+Indeed, there are several implementations of entire vector graphics systems using OpenGL:
\begin{itemize}
-\item over
-\item in
-\item out
-\item atop
-\item xor
-\item plus
+ \item The OpenVG standard\cite{robart2009openvg} has been implemented on top of OpenGL ES\cite{oh2007implementation};
+ \item the Cairo\cite{worth2003xr} library, based around the PostScript/PDF rendering model, has the ``Glitz'' OpenGL backend\cite{nilsson2004glitz}
+ \item and the SVG/PostScript GPU renderer by nVidia\cite{kilgard2012gpu} as an OpenGL extension\cite{kilgard300programming}.
\end{itemize}
-The paper further provides some additional operations for implementing fades and
-dissolves, as well as for changing the opacity of individual elements in a
-scene.
-
-The method outlined in this paper is still the standard system for compositing
-and is implemented almost exactly by modern graphics APIs such as \texttt{OpenGL}. It is
-all but guaranteed that this is the method we will be using for compositing
-document elements in our project.
-
-\subsection{Bresenham's Algorithm: Algorithm for computer control of a digital plotter\cite{bresenham1965algorithm}}
-Bresenham's line drawing algorithm is a fast, high quality line rasterization
-algorithm which is still the basis for most (aliased) line drawing today. The
-paper, while originally written to describe how to control a particular plotter,
-is uniquely suited to rasterizing lines for display on a pixel grid.
-
-Lines drawn with Bresenham's algorithm must begin and end at integer pixel
-coordinates, though one can round or truncate the fractional part. In order to
-avoid multiplication or division in the algorithm's inner loop,
-The algorithm works by scanning along the long axis of the line, moving along
-the short axis when the error along that axis exceeds 0.5px. Because error
-accumulates linearly, this can be achieved by simply adding the per-pixel
-error (equal to (short axis/long axis)) until it exceeds 0.5, then incrementing
-the position along the short axis and subtracting 1 from the error accumulator.
-
-As this requires nothing but addition, it is very fast, particularly on the
-older CPUs used in Bresenham's time. Modern graphics systems will often use Wu's
-line-drawing algorithm instead, as it produces antialiased lines, taking
-sub-pixel coverage into account. Bresenham himself extended this algorithm to
-produce Bresenham's circle algorithm. The principles behind the algorithm have
-also been used to rasterize other shapes, including B\'{e}zier curves.
-
-\emph{GPU Rendering}\cite{loop2005resolution}, OpenVG implementation on GLES: \cite{oh2007implementation},
-\cite{robart2009openvg}
-
-\emph{Existing implementations of document format rendering}
-
-\subsection{Xr: Cross-device Rendering for Vector Graphics\cite{worth2003xr}}
-
-Xr (now known as Cairo) is an implementation of the PDF v1.4 rendering model,
-independent of the PDF or PostScript file formats, and is now widely used
-as a rendering API. In this paper, Worth and Packard describe the PDF v1.4 rendering
-model, and their PostScript-derived API for it.
-
-The PDF v1.4 rendering model is based on the original PostScript model, based around
-a set of \emph{paths} (and other objects, such as raster images) each made up of lines
-and B\'{e}zier curves, which are transformed by the ``Current Transformation Matrix.''
-Paths can be \emph{filled} in a number of ways, allowing for different handling of self-intersecting
-paths, or can have their outlines \emph{stroked}.
-Furthermore, paths can be painted with RGB colours and/or patterns derived from either
-previously rendered objects or external raster images.
-PDF v1.4 extends this to provide, amongst other features, support for layering paths and
-objects using Porter-Duff compositing\cite{porter1984compositing}, giving each painted path
-the option of having an $\alpha$ value and a choice of any of the Porter-Duff compositing
-methods.
-
-The Cairo library approximates the rendering of some objects (particularly curved objects
-such as splines) with a set of polygons. An \texttt{XrSetTolerance} function allows the user
-of the library to set an upper bound on the approximation error in fractions of device pixels,
-providing a trade-off between rendering quality and performance. The library developers found
-that setting the tolerance to greater than $0.1$ device pixels resulted in errors visible to the
-user.
-
-\subsection{Glitz: Hardware Accelerated Image Compositing using OpenGL\cite{nilsson2004glitz}}
-
-This paper describes the implementation of an \texttt{OpenGL} based rendering backend for
-the \texttt{Cairo} library.
-
-The paper describes how OpenGL's Porter-Duff compositing is easily suited to the Cairo/PDF v1.4
-rendering model. Similarly, traditional OpenGL (pre-version 3.0 core) support a matrix stack
-of the same form as Cairo.
-
-The ``Glitz'' backend will emulate support for tiled, non-power-of-two patterns/textures if
-the hardware does not support it.
-
-Glitz can render both triangles and trapezoids (which are formed from pairs of triangles).
-However, it cannot guarantee that the rasterization is pixel-precise, as OpenGL does not proveide
-this consistently.
-
-Glitz also supports multi-sample anti-aliasing, convolution filters for raster image reads (implemented
-with shaders).
+\section{Numeric formats}
+
+On modern computer architectures, there are two basic number formats supported:
+fixed-width integers and \emph{floating-point} numbers. Typically, computers
+natively support integers of up to 64 bits, capable of representing all integers
+between $0$ and $2^{64} - 1$, inclusive\footnote{Most machines also support \emph{signed} integers,
+which have the same cardinality as their \emph{unsigned} counterparts, but which
+represent integers in the range $[-(2^{63}), 2^{63} - 1]$}.
+
+By introducing a fractional component (analogous to a decimal point), we can convert
+integers to \emph{fixed-point} numbers, which have a more limited range, but a fixed, greater
+precision. For example, a number in 4.4 fixed-point format would have four bits representing the integer
+component, and four bits representing the fractional component:
+\begin{equation}
+ \underbrace{0101}_\text{integer component}.\underbrace{1100}_\text{fractional component} = 5.75
+\end{equation}
+
+
+Floating-point numbers\cite{goldberg1992thedesign} are the binary equivalent of scientific notation:
+each number consisting of an exponent ($e$) and a mantissa ($m$) such that a number is given by
+\begin{equation}
+ n = 2^{e} \times m
+\end{equation}
+
+The IEEE 754 standard\cite{ieee754std1985} defines several floating-point data types
+which are used\footnote{Many systems' implement the IEEE 754 standard's storage formats,
+but do not implement arithmetic operations in accordance with this standard.} by most
+computer systems. The standard defines 32-bit (8-bit exponent, 23-bit mantissa, 1 sign bit) and
+64-bit (11-bit exponent, 53-bit mantissa, 1 sign bit) formats\footnote{The 2008
+revision to this standard\cite{ieee754std2008} adds some additional formats, but is
+less widely supported in hardware.}, which can store approximately 7 and 15 decimal digits
+of precision respectively.
+
+Floating-point numbers behave quite differently to integers or fixed-point numbers, as
+the representable numbers are not evenly distributed. Large numbers are stored to a lesser
+precision than numbers close to zero. This can present problems in documents when zooming in
+on objects far from the origin.
+
+IEEE floating-point has some interesting features as well, including values for negative zero,
+positive and negative infinity, the ``Not a Number'' (NaN) value and \emph{denormal} values, which
+trade precision for range when dealing with very small numbers. Indeed, with these values,
+IEEE 754 floating-point equality does not form an equivalence relation, which can cause issues
+when not considered carefully.\cite{goldberg1991whatevery}
+
+There also exist formats for storing numbers with arbitrary precising and/or range.
+Some programming languages support ``big integer''\cite{java_bigint} types which can
+represent any integer that can fit in the system's memory. Similarly, there are
+arbitrary-precision floating-point data types\cite{java_bigdecimal}\cite{boost_multiprecision}
+which can represent any number of the form
+\begin{equation}
+ \frac{n}{2^d} \; \; \; \; n,d \in \mathbb{Z} % This spacing is horrible, and I should be ashamed.
+\end{equation}
+These types are typically built from several native data types such as integers and floats,
+paired with custom routines implementing arithmetic primitives.\cite{priest1991algorithms}
+These, therefore, are likely slower than the native types they are built on.
+
+Pairs of integers $(a \in \mathbb{Z},b \in \mathbb{Z}\setminus 0)$ can be used to represent rationals. This allows
+values such as $\frac{1}{3}$ to be represented exactly, whereas in fixed or floating-point formats,
+this would have a recurring representation:
+\begin{equation}
+ \underbrace{0}_\text{integer part} . \underbrace{01}_\text{recurring part} 01 \; \; 01 \; \; 01 \dots
+\end{equation}
+Whereas with a rational type, this is simply $\frac{1}{3}$.
+Rationals do not have a unique representation for each value, typically the reduced fraction is used
+as a characteristic element.
+
+While traditionally, GPUs have supported some approximation of IEEE 754's 32-bit floats,
+modern graphics processors also support 16-bit\cite{nv_half_float} and 64-bit\cite{arb_gpu_shader_fp64}
+IEEE floats, though some features of IEEE floats, like denormals and NaNs are not always supported.
+Note, however, that some parts of the GPU are only able to use some formats,
+so precision will likely be truncated at some point before display.
+Higher precision numeric types can be implemented or used on the GPU, but are
+slow.\cite{emmart2010high}
-Performance was much improved over the software rasterization and over XRender accellerated rendering
-on all except nVidia hardware. However, nVidia's XRender implementation did slow down significantly when
-some transformations were applied.
+\section{Document Formats}
+Most existing document formats --- such as the venerable PostScript and PDF --- are, however, designed to imitate
+existing paper documents, largely to allow for easy printing. In order to truly take advantage of the possibilities operating in the digital
+domain opens up to us, we must look to new formats.
-\textbf{Also look at \texttt{NV\_path\_rendering}} \cite{kilgard2012gpu}
+Formats such as \texttt{HTML} allow for a greater scope of interactivity and for a more data-driven model, allowing
+the content of the document to be explored in ways that perhaps the author had not anticipated.\cite{hayes2012pixels}
+However, these data-driven formats typically do not support fixed layouts, and the display differs from renderer to
+renderer.
-\section{Floating-Point Precision}
+\subsection{A Taxonomy of Document formats}
+
+The process of creating and displaying a document is a rather universal one (\ref{documenttimeline}), though
+different document formats approach it slightly differently. A document often begins as raw content: text and images
+(be they raster or vector) and it must end up as a stream of photons flying towards the reader's eyes.
+
+\begin{figure}
+ \label{documenttimeline}
+ \centering \includegraphics[width=0.8\linewidth]{figures/documenttimeline}
+ \caption{The lifecycle of a document}
+\end{figure}
+
+There are two fundamental stages by which all documents --- digital or otherwise --- are produced and displayed:
+\emph{layout} and \emph{rendering}. The \emph{layout} stage is where the positions and sizes of text and other graphics are
+determined. The text will be \emph{flowed} around graphics, the positions of individual glyphs will be placed, ensuring
+that there is no undesired overlap and that everything will fit on the page or screen.
+
+The \emph{display} stage actually produces the final output, whether as ink on paper or pixels on a computer monitor.
+Each graphical element is rasterized and composited into a single image of the target resolution.
+
+
+Different document formats cover documents in different stages of this project. Bitmapped images,
+for example, would represent the output of the final stage of the process, whereas markup languages typically specify
+a document which has not yet been processed, ready for the layout stage.
+
+Furthermore, some document formats treat the document as a program, written in
+a (usually turing complete) document language with instructions which emit shapes to be displayed. These shapes are either displayed
+immediately, as in PostScript, or stored in another file, such as with \TeX or \LaTeX, which emit a \texttt{DVI} file. Most other
+forms of document use a \emph{Document Object Model}, being a list or tree of objects to be rendered. \texttt{DVI}, \texttt{PDF},
+\texttt{HTML}\footnote{Some of these formats --- most notably \texttt{HTML} --- implement a scripting lanugage such as JavaScript,
+which permit the DOM to be modified while the document is being viewed.} and SVG\cite{svg2011-1.1}. Of these, only \texttt{HTML} and \TeX typically
+store documents in pre-layout stages, whereas even turing complete document formats such as PostScript typically encode documents
+which already have their elements placed.
+
+\begin{description}
+ \item[\TeX \, and \LaTeX]
+ Donald Knuth's typesetting language \TeX \, is one of the older computer typesetting systems, originally conceived in 1977\cite{texdraft}.
+ It implements a turing-complete language and is human-readable and writable, and is still popular
+ due to its excellent support for typesetting mathematics.
+ \TeX only implements the ``layout'' stage of document display, and produces a typeset file,
+ traditionally in \texttt{DVI} format, though modern implementations will often target \texttt{PDF} instead.
+
+ This document was prepared in \LaTeXe.
+
+ \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[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.
+
+ \item[SVG]
+ Scalable Vector Graphics (SVG) is a vector graphics document format\cite{svg2011-1.1} which uses the Document Object Model. It consists of a tree of matrix transforms,
+ with objects such as vector paths (made up of B\'ezier curves) and text at the leaves.
+
+\end{description}
+
+\subsection{Precision in Document Formats}
+
+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.
-How floating-point works and what its behaviour is w/r/t range and precision
-\cite{goldberg1991whatevery}
-\cite{goldberg1992thedesign}
+\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.
-Arb. precision exists
+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.
-Higher precision numeric types can be implemented or used on the GPU, but are
-slow.
-\cite{emmart2010high}
+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.
+The SVG specification\cite{svg2011-1.1} specifies numbers as strings with a decimal representation of the number.
+It is stated that a ``Conforming SVG Viewer'' must have ``all visual rendering accurate to within one device pixel to the mathematically correct result at the initial 1:1
+zoom ratio'' and that ``it is suggested that viewers attempt to keep a high degree of accuracy when zooming.''
+A ``Conforming High-Quality SVG Viewer'' must use ``double-precision floating point\footnote{Presumably the 64-bit IEEE 754 ``double'' type.}'' for computations involving
+coordinate system transformations.
\section{Quadtrees}
-The quadtree is a data structure which
-\cite{finkel1974quad}
-
+When viewing or processing a small part of a large document, it may be helpful to
+only processs --- or \emph{cull} --- parts of the document which are not on-screen.
+
+\begin{figure}[h]
+ \centering \includegraphics[width=0.4\linewidth]{figures/quadtree_example}
+ \caption{A simple quadtree.}
+\end{figure}
+The quadtree\cite{finkel1974quad}is a data structure --- one of a family of \emph{spatial}
+data structures --- which recursively breaks down space into smaller subregions
+which can be processed independently. Points (or other objects) are added to a single
+node which (if certain criteria are met) is split into four equal-sized subregions, and
+points attached to the region which contains them.
+
+Quadtrees have been used in computer graphics for both culling --- excluding objects in
+nodes which are not visible --- and ``level of detail'', where different levels of the quadtree store
+different quality versions of objects or data\cite{zerbst2004game}.
+Typically the number of points in a node
+exceeding a maximum triggers this split, though in our case likely the level of precision required exceeding
+that supported by the data type in use.
+
+In this project, we will be experimenting with a form of quadtree in which each
+node has its own independent coordinate system, allowing us to store some spatial
+information\footnote{One bit per-coordinate, per-level of the quadtree} within the
+quadtree structure, eliminating redundancy in the coordinates of nearby objects.
+
+Other spatial data structures exist, such as the KD-tree\cite{bentley1975multidimensional},
+which partitions the space on any axis-aligned line; or the BSP tree\cite{fuchs1980onvisible},
+which splits along an arbitrary line which need not be axis aligned. We believe, however,
+that the simpler conversion from binary coordinates to the quadtree's binary split make
+it a better avenue for initial research to explore.
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