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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
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% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
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% Document Info
\author{racheleva}
\pdfinfo{
  /Title (computer-graphics-4.pdf)
  /Creator (Cheatography)
  /Author (racheleva)
  /Subject (Computer Graphics 4 Cheat Sheet)
}

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% Header and Footer
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\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 4 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{racheleva} via \textcolor{DarkBackground}{\uline{cheatography.com/217931/cs/48019/}}}
\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 28th April, 2026.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \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

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% 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{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{UNIT 4}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{PART A} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{modeling transformations}} & {\bf{Viewing transformations}} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
world coordinate system. & re-position the scene in front of a camera \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
{\bf{view orientation transformation}} & DC to WC; camera at origin looking along -z axis \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
{\bf{parallel projection}} & {\bf{perspective projection}} \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
preserves true measurements; less realistic & realistic (depth); does not preserve true measurements \tn 
% Row Count 14 (+ 3)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Center of Projection}} (COP) or {\bf{Projection Reference Point}} (PRP) is the single point where all projectors converge} \tn 
% Row Count 17 (+ 3)
% Row 7
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Viewport Clipping}} -  after projection in standard viewing pipeline (before the perspective projection division; avoids division by zero)} \tn 
% Row Count 20 (+ 3)
% Row 8
\SetRowColor{LightBackground}
{\bf{Back-Face Detection condition}} & V = N • D \tn 
% Row Count 22 (+ 2)
% Row 9
\SetRowColor{white}
{\bf{Scan-Line Method}} & determines the closest object along the line of sight (depth) and its corresponding color \tn 
% Row Count 27 (+ 5)
% Row 10
\SetRowColor{LightBackground}
{\bf{Painter's Algorithm.}} - Depth-Sorting Method & {\bf{Primary criterion: }}sorted by minimum z-depth\{\{nl\}\}Ambiguity tests: Bounding rectangle overlap test, depth overlap test, surface intersection test, and final "complete behind" test. \tn 
% Row Count 37 (+ 10)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{composite transformation matrix is not commutative \newline limitation of Back-Face \newline Detection : multiple front faces overlap or obscure each other (hard to detect)}  \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}{Parallel vs. Perspective projection}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Parallel projection}} & {\bf{Perspective projection}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Orthographic: perpendicular rays \{\{nl\}\}Oblique: angled rays & converge at a point (foreshortening) \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
No depth cues, looks artificial, parallel lines stay parallel & Realistic, depth visible, lines converge (vanishing points) \tn 
% Row Count 9 (+ 4)
% Row 3
\SetRowColor{white}
Technical Drawings & Realistic Visualization \tn 
% Row Count 11 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Parallel → accuracy \& measurement \newline Perspective → realism \& depth}  \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}{Back-Face Detection algorithm}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Removes polygons whose normal faces away from viewer} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{N⋅V≤0⇒discard} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{Sufficient for:}} & Single, convex, closed objects (e.g., cube) \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Limitations}}} \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
{\bf{1. Concave Objects:}} & May remove faces that are actually visible \tn 
% Row Count 10 (+ 3)
% Row 5
\SetRowColor{white}
{\bf{2. Multiple Objects:}} & Cannot decide visibility between different objects; No depth comparison \tn 
% Row Count 14 (+ 4)
% Row 6
\SetRowColor{LightBackground}
{\bf{3. Transparent Surfaces:}} & Incorrectly removes visible back faces \tn 
% Row Count 16 (+ 2)
% Row 7
\SetRowColor{white}
{\bf{4. Open Surfaces:}} & Back-face concept may not apply \tn 
% Row Count 18 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{5. Does not resolve cyclic overlap cases} \tn 
% Row Count 19 (+ 1)
% Row 9
\SetRowColor{white}
6. Object-space only; doesn't compare between objects & does not work for scene-level visibility \tn 
% Row Count 22 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{It is a pre-processing culling step that reduces the polygon count sent to the rendering pipeline (by \textasciitilde{}50\%). \newline followed by a true {\bf{Hidden Surface Removal }}(HSR) algorithm (e.g., Depth-Buffer/Z-Buffer, Painter's, BSP Trees)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{3.38436 cm} x{1.59264 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{3D clipping}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Steps}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{1. Transform vertices → Normalized Device Coordinates (NDC) using projection} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
2. Clip against 6 planes of canonical cube: & x=±1, y=±1, z=0/1 \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{3. Test vertices (inside/outside)} \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{4. Compute intersection points if edges cross planes} \tn 
% Row Count 8 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{5. Discard outside parts, pass clipped primitives for viewport mapping} \tn 
% Row Count 10 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Standardized Cube}}} \tn 
% Row Count 11 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{Uniform planes (inside/out and intersection math faster); hardware optimized; simplified depth (0 to 1)} \tn 
% Row Count 14 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Clipping in normalized space:}} consistency \& simplicity (axis-aligned with plane equations than skewed, perspective dependent frustum); device independence; efficiency}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.04057 cm} x{2.93643 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{3D projection}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Perspective Projection}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
1. find parameter t & t=z\textasciitilde{}p\textasciitilde{}-z\textasciitilde{}CoP\textasciitilde{}/z-z\textasciitilde{}CoP \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
2. projected x' and y' & x′=x\textasciitilde{}CoP\textasciitilde{}​+t(x−x\textasciitilde{}CoP\textasciitilde{}​)\{\{nl\}\}y′=y\textasciitilde{}CoP\textasciitilde{}​+t(y−y\textasciitilde{}CoP​\textasciitilde{}) \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
3. projected z' & z′=z\textasciitilde{}p\textasciitilde{}​ \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Orthographic Projection}}} \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
1. Keep x and y unchanged & x′=x, y′=y \tn 
% Row Count 10 (+ 2)
% Row 6
\SetRowColor{LightBackground}
2. Set z to projection plane & z′=z\textasciitilde{}p​\textasciitilde{} \tn 
% Row Count 12 (+ 2)
\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}{Multi-Viewport Viewing Pipeline for CAD}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{1. Divergence of Single Modeling Stream}} & One modeling transformation pipeline (world/model matrix) feeds all viewports. \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
{\bf{2. Viewing \& Projection Management}} & Each viewport has its own:  \{\{nl\}\}{\bf{View Matrix}} (camera orientation)\{\{nl\}\}{\bf{Projection Matrix}}\{\{nl\}\} Orthographic (Top, Front, Side) \{\{nl\}\}Perspective (3D view) \tn 
% Row Count 13 (+ 9)
% Row 2
\SetRowColor{LightBackground}
{\bf{3. Clipping per Viewport}} & Clipping done independently for each viewport (frustum) \tn 
% Row Count 16 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{4. Overall architecture}}} \tn 
% Row Count 17 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Model Transform → Stream Duplication → Parallel Per-Viewport (View+Proj Transform → Clipping → Viewport Transform).} \tn 
% Row Count 20 (+ 3)
\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}{Multi-Viewport Viewing Pipeline for CAD}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{1. Divergence of Single Modeling Stream}} & One modeling transformation pipeline (world/model matrix) feeds all viewports. \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
{\bf{2. Viewing \& Projection Management}} & Each viewport has its own:  \{\{nl\}\}{\bf{View Matrix}} (camera orientation)\{\{nl\}\}{\bf{Projection Matrix}}\{\{nl\}\} Orthographic (Top, Front, Side) \{\{nl\}\}Perspective (3D view) \tn 
% Row Count 13 (+ 9)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}