+\documentclass[a4paper,10pt]{cshonours}
+\bibliographystyle{acm}
+\usepackage[utf8]{inputenc}
+\usepackage{hyperref}
+\usepackage{graphicx}
+\usepackage{lastpage}
+\usepackage{fancyhdr}
+%\usepackage{listing}
+\usepackage{subcaption}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{minted}
+\pagestyle{fancyplain}
+
+%opening
+\title{Precision in vector documents: a spatial approach}
+\author{David Gow}
+
+\begin{document}
+\cfoot{}
+\lfoot{\thepage\ of \pageref{LastPage}}
+\rfoot{DRAFT}
+\rhead{David Gow (20513694)}
+\lhead{\today}
+
+\maketitle
+
+\tableofcontents
+
+\chapter{Introduction}
+
+Documents are an important part of day-to-day life: we use them for business and pleasure,
+to inform and entertain. We often use images or diagrams to illustrate our ideas, and to convey
+spatial or visual information.
+
+On the printed page, there is a practical limit to the size and resolution of these images.
+In order to store, for example, maps of large areas with fine detail, we resort to an atlas:
+splitting the map up into several pages, each of which covers a different part of the map.
+
+With the advent of computers, we can begin to alleviate these problems. The fixed size and resolution
+of the screen is worked around by having the screen be a \emph{viewport} into a larger document, letting
+the document be \emph{panned} (translated laterally) and \emph{zoomed} (scaled to a particular magnification).
+
+However, limits in the range and precision of the numbers used to store and manipulate documents
+artificially restrict the size and zoom-level of documents. While changing the numeric types used
+can solve these issues, here we investigate the use of a quadtree to take advantage of the spatial
+nature of the data. This is akin to maintaining the atlas of several different pages with different views
+of the map, but with the advantage of greater ease-of-use and continuity.
+
+\chapter{Background}
+
+\section{Rendering}
+
+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 match how cameras, printers and monitors work.
+
+
+\begin{figure}[h]
+ \centering \includegraphics[width=0.8\linewidth]{figures/vectorraster_example}
+ \caption{\label{vectorraster}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 (see Figure \ref{vectorraster}) 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 avoid many of these problems: the representation is independent of the output resolution, and rather
+an abstract description of what is being rendered, typically as a combination of simple geometric shapes like lines,
+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. The resulting bitmap is then an approximation of the vector image
+at that resolution. This bitmap is then displayed or printed.
+
+% Project specific line
+We will discuss the use of vector graphics, as they lend themselves more naturally to scaling,
+though many of the techniques will also apply to raster graphics. Indeed, the use of quadtrees
+to compress bitmapped data is well established\cite{salomon2007data}, and the view-reparenting
+techniques have been used to facilitate an infinite zoom into a (procedurally generated) document\cite{munroe2014pixels}.
+
+\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.
+
+\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 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}
+
+
+\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, taocp2} are a superset of scientific notation,
+originally used by the Babylonians (in base 60) perhaps as early as 1750 BC.
+Each number $n$ (in a base $\beta$) consists of integers $e$ (the \emph{exponent}) and $m$ (the \emph{mantissa}) such that
+\begin{equation}
+ n = \beta^{e} \times m_f
+\end{equation}
+
+Both $e$ and $m$ typically have a fixed width $q$ and $p$ respectively in base $\beta$. For notational convenience, we also
+represent $m$ as a fraction $m_f = \beta^{-p} m$ so $|m_f| < 1$. We further call a floating point number
+\emph{normalised} if the first digit of $m$ is nonzero. The exponent $e$ is usually allowed to be
+either positive or negative (either by having a sign bit, or being offset a fixed amount). The value
+of a floating point number $n$ must therefore be $-\beta^{e_max} < n < \beta^{e_max}$. The smallest
+possible magnitude of a normalised floating point number is $\beta^{e_min-1}$, as $m_f$ must be at least $0.1$.
+Non-normalised numbers may represent $0$, and many normalised floating-point include zero in some way, too.
+
+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\cite{openglspec, ARBshaderprecision}.} 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. The IEEE 754 standard also introduces an implicit $1$ bit in the most
+significant place of the mantissa when the exponent is not $e_min$.
+
+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. Furthermore, due to the limited precision, and the different
+``alignment'' of operands in arithmetic, several algebraic properties of rings we take for
+granted in the integers do not exist, including the property of assosciativity\cite{taocp2}.
+
+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{subnormal} values, which
+trade precision for range when dealing with very small numbers by not normalising
+numbers when the exponent is $e_min$. 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}
+Operations on these types, therefore, are usually slower than those performed on native types.
+
+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 not able to use all 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}
+
+
+\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.
+
+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.
+
+\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{\label{doclifecycle}The lifecycle of a document}
+\end{figure}
+
+There are two fundamental stages (as shown in Figure \ref{doclifecycle}) 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.
+
+\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
+$\pm1.175 \times 10^{-38}$ with approximately $5$ decimal digits of precision \emph{in the fractional part}.
+\footnote{The PDF specification mistakenly leaves out the negative in the exponent here.}
+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}
+When viewing or processing a small part of a large document, it may be helpful to
+only process --- 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.
+
+\chapter{\texttt{IPDF}: A Document Precision Playground}
+In order to investigate the precision issues present in document formats and viewers,
+we developed a document viewer \texttt{IPDF}. \texttt{IPDF} is a \texttt{C++} program
+which supports rendering TrueType font outlines and a subset of the \texttt{SVG}
+format.
+
+At its core, \texttt{IPDF} breaks documents down into a set of \emph{objects}, each
+with rectangular bounds $(x, y, w, h)$ and with several \emph{types}:
+\begin{enumerate}
+ \item \textbf{Rectangle:} A simple, axis-aligned rectangle (either filled with a solid colour or
+ left as an outline) covering the object bounds.
+ \item \textbf{Ellipse:} A filled, axis-aligned ellipse, rendered parametrically.
+ \item \textbf{B\'ezier Curve:} A cubic B\'ezier curve. B\'ezier curve
+ control points are stored relative to the coordinate system provided by the
+ document bounds, and several objects can share these coefficients.
+\end{enumerate}
+
+\texttt{IPDF} can be compiled using different number representations, to allow for
+comparison between not only different-precision floating point numbers, but also
+arbitrary-precision types such as rationals.
+
+Documents in \texttt{IPDF} may be rendered either on the CPU, using a custom software
+rasterizer built on the numeric types supported, or on the GPU with OpenGL 3.2 shaders, using
+the GPU's default numeric representation.
+
+Furthermore, \texttt{IPDF} allows \texttt{SVG} files (and text rendered with TrueType fonts) to be inserted at any point and scale in
+the document, allowing for the same images to be compared at different scales.
+
+\section{GPU Floating Point Rounding Behaviour}
+While the IEEE-754 standard specifies both the format of floating-point numbers and the operations they perform.
+However, the OpenGL specification\cite{openglspec}, while requiring the same basic format for single-precision floats,
+does not specify the behaviour of denormals, nor requires any support for NaN or infinite values. Similarly, no support
+for floating point exceptions is required, with the note that no operation should ever halt the GPU.
+
+However, an extension to the specification, \texttt{GL\_ARB\_shader\_precision}\cite{ARBshaderprecision} in 2010
+allows programs to require stricter precision requirements. Notably, support for infinite values is required and
+maximum relative error in ULPs.
+
+Different GPU vendors and drivers have different rounding behaviour, and most hardware (both CPU and GPU) provide
+ways of disabling support for some IEEE features to improve performance\cite{intelgpuspec, dawson2012notnormal}.
+
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.8\textwidth]{figures/circles_gpu_vs_cpu/comparison.pdf}
+\caption{\label{fig:circles}The edges of a unit circle viewed through bounds (x,y,w,h) = (0.0869386,0.634194,2.63295e-07,2.63295e-07)}
+\end{figure}
+
+Many GPUs now support double-precision floating-point numbers\footnote{Some drivers, however, do not yet support this feature, including one of our test machines.}
+as specified in the \texttt{GL\_ARB\_gpu\_shader\_fp64}\cite{arb_gpu_shader_fp64} OpenGL extension.
+However, many of the fixed-function parts of the GPU do not support double-precision floats,
+making it impractical to use them.
+
+To see the issues, we rendered the edge of a circle (calculated by discarding pixels with $x^2 + y^2 > 1$) on several GPUs, as well as an \texttt{x86-64} CPU,
+as seen in \autoref{fig:circles}.
+Of these, the nVidia GPU came closest to the CPU rendering, whereas Intel's hardware clearly performs some optimisation which produces quite different artefacts.
+The diagonal distortion in the AMD rendering may be a result of different rounding across the two triangles across which the circle was rendered.
+\section{Distortion and Quantisation at the limits of precision}
+TODO: Fix these figures, explain properly.
+\begin{figure}[h]
+
+ \begin{subfigure}{.5\linewidth}
+ \includegraphics[width=\linewidth]{cpu0}
+ \subcaption{Intended rendering}
+ \end{subfigure}
+ \begin{subfigure}{.5\linewidth}
+ \includegraphics[width=\linewidth]{cpu100}
+ \subcaption{With artefacts}
+ \end{subfigure}
+ \caption{\label{fig:precisionerrors}Floating point errors affecting the rendering of an image}
+
+\end{figure}
+
+When trying to insert fine detail into a document using fixed-width floats, some precision is lost. In particular, the
+control points of the curves making up the image get rounded to the nearest\footnote{Other rounding modes are available,
+but they all suffer from similar artefacts.} representable point. \autoref{fig:precisionerrors} shows what happens when an
+image is small enough to be affected by this quantisation. The grid structure becomes very apparent.
+
+These artefacts also become prevalent when an object is far from the origin, as more bits would be required to store the
+position, so the lower order bits of the position must be discarded. As the position of the viewport and the object will share
+many of the same initial digits, catastrophic cancellation\cite{goldberg1991whatevery} can occur.
+
+%\subsection{Big Integers}
+%Computers typically operate on \emph{fixed-width} integers\footnote{The term ``machine word'' has traditionally been used
+%for a CPUs default numeric type, though for historical reasons this is sometimes also used to refer to a 16-bit type, regardless
+%of the preferred type of the underlying machine.} for which they have hardware adders.
+
+%These types are represented by $n$ bits (typically $n$ is 32 or 64), and represents the
+%ring $\mathbb{Z}_n$: i.e.\ any number in the range $[0,2^{n})$ can be represented, and operations
+%are performed $\mod 2^n$.
+\chapter{View Reparenting and the Quadtree}
+One of the important parts of document rendering is that of coordinate transforms.
+
+Traditionally, the document exists in its own coordinate system, $\mathbf{b}$.
+We then define a coordinate system $\mathbf{v}$ to represent the view.
+
+To eliminate visible error due to floating point precision, we need to ensure that
+any point (including control points) must be representable as a float both in the
+document and in view coordinates, i.e:
+\begin{enumerate}
+ \item have a limited magnitude,
+ \item be precise enough to uniquely identify a pixel on the display.
+\end{enumerate}
+
+Despite a point not being representable with floats in one coordinate system,
+it may be representable in another. We can therefore split a document up into
+several coordinate systems, such that each point is completely representable.
+
+Similarly, the points making up the visible document need to all be representable
+(to at least one pixel's worth of precision). To achieve this, we use a quadtree where
+each node stores points within its bounds in its own coordinate system. Objects
+which span multiple nodes are clipped, such that no points\footnote{except some
+control points} lie outside the quadtree node.
+
+Setting aside the possibility that an object might span multiple nodes for the time
+being, let's investigate how the coordinates of a point are affected by placing it
+in the quadtree.
+
+Suppose $p = (p_x, p_y)$ is a point in a global document coordinate system, which we'll
+consider the root node of our quadtree. $p_x$ and $p_y$ are represented by a finite
+list of binary digits $x_0 \dots x_n$, $y_0 \dots y_n$.
+Consider now the pair $(x_0, y_0)$:
+\begin{align*}
+ x_0 &= \begin{cases}
+ 0 & \text{if }p_x\text{ is in the left half of }d\\
+ 1 & \text{if }p_x\text{ is in the right half of }d
+ \end{cases} \\
+ y_0 &= \begin{cases}
+ 0 & \text{if }p_y\text{ is in the bottom half of }d\\
+ 1 & \text{if }p_y\text{ is in the top half of }d
+ \end{cases}
+\end{align*}
+
+We have therefore found which child node of our quadtree contains $p$. The coordinates
+of $p$ within that node are $(x_1 \dots x_n, y_1 \dots y_n)$. By the principle of mathematical
+induction, we can repeat the process to move more bits from the coordinates of $p$ into the
+structure of the quadtree, until the remaining coordinates may be precisely represented.
+
+This implies that the quadtree is equivalent to an arbitrary precision integer datatype. By reversing
+the process, any point representable in the quadtree can be stored as a fixed-length bit string.
+
+%It is natural to treat these coordinates as being between $0$ and $1$.
+
+Furthermore, we can get approximations of points by inserting them into higher levels of the quadtree,
+with their coordinates rounded. By then viewing the document at a certain level of the quadtree, we
+then not only do not need to consider bits of precision which are not required to represent the point to
+pixel resolution, we can also cull objects outside the visible nodes at that level.
+
+Indeed, we can guarantee that, by selecting the level of the quadtree we view such that
+the view width and height lie within the range $(0.5,1]$, we can ensure that at most
+four nodes need to be rendered.
+
+
+\section{Clipping cubic B\'eziers}
+In order to ensure that the quadtree maintains precision in this fashion, we need to ensure
+that all objects are entirely contained within a quadtree node. This is not always the case and,
+indeed, when zooming in on any object, eventually the object will span multiple quadtree nodes.
+
+{\bf TODO: We can show this by showing that, given $(a,b) \in \mathbb{R}, \exists $ binary fraction $c$ such that $a < c < b$.}
+
+We therefore need a way to subdivide objects into several objects, each of which is contained within one node.
+To do this, we need to \emph{clip} the cubic B\'ezier curves from which our document is formed to a rectangle.
+
+Cubic B\'ezier curves are defined parametrically as two cubics: $x(t)$ and $y(t)$, with $0 \leq t \leq 1$.
+We clip these to a rectangular bounding box in stages. We first find the intersections of the curve with
+the clipping rectangle by finding the roots\footnote{While there is a method for solving cubics exactly\cite{cardano1545artis},
+we instead use numeric methods to avoid the need for square root operations, which cannot be done exactly
+on some of the numeric types we used.}
+of the cubic shifted to match the corners of the rectangle. This produces some spurious points, as it assumes the edges
+of the clipping rectangle are lines with infinite extent, but this at worst introduces some minor inefficiency in the process and does
+not affect the result.
+
+Once the values of the parameter $t$ which intersect the clipping rectangle have
+been determined, the curve is split into segments. To do this, the values $0$ and $1$
+(representing the endpoints) are added to the list of intersecting $t$ values, which is then
+sorted. Adjacent values $t_0$ and $t_1$ form a segment. The midpoint of that segment (with the
+value $\frac{t_0 + t_1}{2}$) is evaluate and if it falls
+outside the clipping rectangle, the segment is discarded.
+
+Finally, the segments are re-parametrised by subdividing the curve using De Casteljau's
+algorithm\cite{sederberg2007cad}. By re-parameterising the curves such that $0 \leq t \leq 1$,
+we ensure that the first and last coefficients have the endpoints' coodinates, and therefore
+lie in the quadtree node.
+
+{\bf TODO: Prove that the other control points' magnitude is reduced, and try to quantify it, prove that
+it will never overflow.}
+
+\section{Implementation Details}
+\begin{itemize}
+ \item Store object ID ranges.
+ \item Pointers to children and parent.
+ \item Linked-list of ``overlay'' nodes for mutation.
+ \item Have billions of bugs.
+\end{itemize}
+
+
+
+\chapter{Experimental Results}
+These are all iPython-y at the moment.
+
+Roughly 3s/frame for GMP rationals, 16ms for Quadtree which is still slightly broken.
+\section{Performance per object}
+
+\section{Performance per onscreen object}
+
+\section{Performance per zoom-level}
+
+\section{Stability of performance}
+
+\chapter{Further Work and Conclusion}
+The use of spatial data structures to specify documents with arbitrary precision
+has been validated as a viable alternative to the use of arbitrary precision numeric
+types where an arbitrary (rather than infinite) amount of precision is needed.
+Once constructed, they are faster in many circumstanced, and the structure
+can also be used for culling. When the viewport moves and scales smoothly, the cost
+of constructing new nodes is amortised over the movement.
+Unfortunately, the mutation of the quadtree is difficult and slow, and discontinuous
+movement can result in a large number of nodes needing to be created.
+
+Quadtree seems to be viable and is really performant.
+
+Loop-blinn shading.
+
+
+\bibliography{papers}
+
+\end{document}
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