\documentclass[10pt,a4paper]{article}

% Packages
\usepackage{fancyhdr}           % For header and footer
\usepackage{multicol}           % Allows multicols in tables
\usepackage{tabularx}           % Intelligent column widths
\usepackage{tabulary}           % Used in header and footer
\usepackage{hhline}             % Border under tables
\usepackage{graphicx}           % For images
\usepackage{xcolor}             % For hex colours
%\usepackage[utf8x]{inputenc}    % For unicode character support
\usepackage[T1]{fontenc}        % Without this we get weird character replacements
\usepackage{colortbl}           % For coloured tables
\usepackage{setspace}           % For line height
\usepackage{lastpage}           % Needed for total page number
\usepackage{seqsplit}           % Splits long words.
%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{racheleva}
\pdfinfo{
  /Title (computer-graphics-2.pdf)
  /Creator (Cheatography)
  /Author (racheleva)
  /Subject (Computer Graphics 2 Cheat Sheet)
}

% Lengths and widths
\addtolength{\textwidth}{6cm}
\addtolength{\textheight}{-1cm}
\addtolength{\hoffset}{-3cm}
\addtolength{\voffset}{-2cm}
\setlength{\tabcolsep}{0.2cm} % Space between columns
\setlength{\headsep}{-12pt} % Reduce space between header and content
\setlength{\headheight}{85pt} % If less, LaTeX automatically increases it
\renewcommand{\footrulewidth}{0pt} % Remove footer line
\renewcommand{\headrulewidth}{0pt} % Remove header line
\renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit
% This two commands together give roughly
% the right line height in the tables
\renewcommand{\arraystretch}{1.3}
\onehalfspacing

% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
\newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols
\newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns
\newcommand{\tn}{\tabularnewline} % Required as custom column type in use

% Font and Colours
\definecolor{HeadBackground}{HTML}{333333}
\definecolor{FootBackground}{HTML}{666666}
\definecolor{TextColor}{HTML}{333333}
\definecolor{DarkBackground}{HTML}{14A35B}
\definecolor{LightBackground}{HTML}{F0F9F4}
\renewcommand{\familydefault}{\sfdefault}
\color{TextColor}

% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Computer Graphics 2 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{racheleva} via \textcolor{DarkBackground}{\uline{cheatography.com/217931/cs/48016/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}racheleva \\
  \uline{cheatography.com/racheleva} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 27th April, 2026.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{UNIT 2}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{PART A} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{homogeneous  coordinates}} - represent all transformations as a uniform matrix multiplication; easy composition} \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{viewing pipeline}} - mapping world coordinate system to viewport (display device)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{modeling coordinates}} are local to the object and {\bf{world  coordinates}} are the common reference system} \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{normalization transformation}} - maps the world coordinate window to a standard, device-independent coordinate system (simplify {\bf{clipping }}and {\bf{viewport mapping}})} \tn 
% Row Count 13 (+ 4)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Sutherland-Hodgman:}}\{\{nl\}\} P1 inside, P2 inside → Output P2\{\{nl\}\} P1 inside, P2 outside → Output Intersection \{\{nl\}\} P1 outside, P2 inside → Output I, then P2 \{\{nl\}\}P1 outside, P2 outside → Output nothing} \tn 
% Row Count 18 (+ 5)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Line clipping algorithms:}} Cohen-Sutherland, Liang-Barsky (or Nicholl–Lee–Nicholl (NLN), Cyrus-Beck} \tn 
% Row Count 21 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{fig. 1}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/racheleva_1777283683_Screenshot 2026-04-27 152400.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Cohen Sutherland}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Transformations}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/racheleva_1777302526_Untitled-2025-06-11-0853.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Homogeneous Coordinates}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Extension of 2D Cartesian coordinates (x, y) to 3D (x, y, w).} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Point represented as a column vector {[}x, y, 1{]}\textasciicircum{}T.} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Without homogeneous coordinates, translation is vector addition (T + v), while others are matrix multiplications (M * v).} \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{1. Unified Representation:}} 3x3 matrix multiplication} \tn 
% Row Count 9 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{2. Efficient Concatenation:}} multiple transformations} \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{3. Elegant Handling of Translation:}} translation requires addition} \tn 
% Row Count 13 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{4. Ability to Represent Points at Infinity:}} advanced graphical concepts} \tn 
% Row Count 15 (+ 2)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{5. Simplified Inverse Transform:}} inverse of the single composite matrix,} \tn 
% Row Count 17 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Trade-off with Cartesian system:}} \newline {\bf{1. Conceptual Complexity}} - understanding an extended coordinate system \newline {\bf{2. Computational Complexity}} - slight increase (3x3 instead of 2x2) per operation; though there is  a computational gain through matrix concatenation (composite matrix) \newline ⇒ composite matrix transformations happen right to left}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.55618 cm} x{1.51041 cm} x{1.51041 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Polygon tables}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Vertex + Polygon Tables}} & {\bf{Single Integrated Polygon Table}} & {\bf{Three-Table Model}} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Vertex table (8 coords) + Polygon table (6 faces) & Each face stores full vertex coordinates (6 faces, 3 coordinates) & Vertex table + Edge table + Polygon table \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
24 floats + 24 ints & 72 floats & 24 floats + 48 ints \tn 
% Row Count 10 (+ 2)
% Row 3
\SetRowColor{white}
efficient, no duplication & simple & full topology, adjacency queries \tn 
% Row Count 13 (+ 3)
% Row 4
\SetRowColor{LightBackground}
no explicit edge info & high memory & more memory + complex \tn 
% Row Count 15 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{3}{x{5.377cm}}{{\bf{Data Efficiency}}} \tn 
% Row Count 16 (+ 1)
% Row 6
\SetRowColor{LightBackground}
best & worst & moderate \tn 
% Row Count 17 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{Integrated = simple but wasteful | Three-table = powerful but heavy | Vertex+Polygon = best balance}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{2D viewing pipeline process}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{1. World Window Definition}} & a rectangular region in the World Coordinate System (clipping boundary) \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
{\bf{2. Viewport Definition}} & a rectangular region on the Device Coordinate System \tn 
% Row Count 7 (+ 3)
% Row 2
\SetRowColor{LightBackground}
{\bf{3. Window-to-Viewport Mapping Process}} & linear transformation (mapping) from window to viewport \tn 
% Row Count 10 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Steps}}} \tn 
% Row Count 11 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{1. Translate window → origin} \tn 
% Row Count 12 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{2. Scale to viewport size} \tn 
% Row Count 13 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{3. Translate to viewport position} \tn 
% Row Count 14 (+ 1)
% Row 7
\SetRowColor{white}
(Additionally) P\textasciitilde{}screen\textasciitilde{}​=M\textasciitilde{}screen\textasciitilde{}​×M\textasciitilde{}viewport​\textasciitilde{}×M\textasciitilde{}zoom\textasciitilde{}​×M\textasciitilde{}pan​\textasciitilde{}×P\textasciitilde{}world​\textasciitilde{} & {\bf{Pan:}} M\textasciitilde{}pan\_i\textasciitilde{}=Translate by (-T\textasciitilde{}xi\textasciitilde{}, -T\textasciitilde{}yi\textasciitilde{}) \{\{nl\}\}{\bf{Zoom:}} M\_zoom\_i = Scale by (S\textasciitilde{}i\textasciitilde{}, S\textasciitilde{}i\textasciitilde{}) \tn 
% Row Count 20 (+ 6)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Flexibility:}} Separating pan \& zoom matrices allows independent control.} \tn 
% Row Count 22 (+ 2)
% Row 9
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Efficiency:}} GPU-friendly single matrix multiplication} \tn 
% Row Count 24 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Combined Importance}} \newline 1. Separation of Concerns (decouples scene design from display specs) \newline 2. Flexibility \& Reuse \newline 3. Device Independence}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.23965 cm} x{2.73735 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Sutherland–Hodgman}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Inside test}} & x\textasciitilde{}min\textasciitilde{}​≤x≤x\textasciitilde{}max\textasciitilde{}​, y\textasciitilde{}min\textasciitilde{}​≤y≤y\textasciitilde{}max​\textasciitilde{} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{computationally cheap} \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{Intersection Checks for Lines}} & point where a polygon edge crosses a clip boundary. \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{Use line equations to calculate intersection coordinates} \tn 
% Row Count 9 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{intersection becomes a new vertex; closed and correctly shaped} \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
{\bf{Iterative Vertex Generation}} & processes the polygon against each clip boundary (left, right, bottom, top) successively. \tn 
% Row Count 16 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Modular and systematic}} - 4 simple clipping steps \newline {\bf{Builds final result incrementally}} }  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.89126 cm} x{3.08574 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Cohen–Sutherland}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Assign Region Codes}} & For both end points P\textasciitilde{}1\textasciitilde{} and P\textasciitilde{}2\textasciitilde{} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Perform Trivial Tests}}} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Trivial Acceptance: \seqsplit{(code1​∣code2​)=0000}} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{Trivial Rejection: \seqsplit{(code1​\&code2​)≠0000}} \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
{\bf{Select Outside Point}} & Choose endpoint with non-zero region code \tn 
% Row Count 7 (+ 2)
% Row 5
\SetRowColor{white}
{\bf{Find Intersection}} & Left/Right boundary (x = constant):  y=y\textasciitilde{}1\textasciitilde{}​+m(x−x\textasciitilde{}1\textasciitilde{})  Top/Bottom boundary (y = constant):  x=x\textasciitilde{}1\textasciitilde{}​+ (y−y\textasciitilde{}1\textasciitilde{}​)/m 	​ \tn 
% Row Count 13 (+ 6)
% Row 6
\SetRowColor{LightBackground}
{\bf{Repeat}} & Continue until: Accepted (both codes = 0000), or Rejected (AND ≠ 0) \tn 
% Row Count 16 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}