tiet-question-paper (23jun24)

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diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/LICENSE b/Master/texmf-dist/doc/latex/tiet-question-paper/LICENSE
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+The MIT License (MIT)
+Copyright (c) 2024 B.V. Raghav
+
+Permission is hereby granted, free of charge, to any
+person obtaining a copy of this software and associated
+documentation files (the ``Software''), to deal in the
+Software without restriction, including without
+limitation the rights to use, copy, modify, merge,
+publish, distribute, sublicense, and/or sell copies of
+the Software, and to permit persons to whom the
+Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice
+shall be included in all copies or substantial portions
+of the Software.
+
+THE SOFTWARE IS PROVIDED ``AS IS'', WITHOUT WARRANTY OF
+ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED
+TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
+PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT
+SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR
+ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
+OTHER DEALINGS IN THE SOFTWARE.
diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/README.md b/Master/texmf-dist/doc/latex/tiet-question-paper/README.md
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+# `tiet-question-paper.cls` #
+
+[Installation](#installation) | 
+[Usage](#usage) | 
+[Emacs Org Integration](#emacs-org-integration)
+
+Author: Raghav B. Venkataramaiyer  
+Email: bv.raghav -at- thapar -dot- edu  
+Email: bvraghav -at- gmail -dot- com  
+License: [MIT License](./LICENSE)  
+Version: 2024-06-22
+
+Provide a document class `tiet-question-paper` in order
+to create a question paper for [the Thapar
+Institute](https://thapar.edu/) in $\LaTeX$.  Although
+created for the TIET, the module, however, is adaptable
+to any organisation.
+
+The following example is a screenshot from
+[`sample.pdf`](./sample.pdf), with the source at
+[`sample.tex`](./sample.tex)
+
+![](./images/sample.png)
+
+## Installation ##
+
+### CTAN ###
+
+TBA
+
+### From Source ###
+
+1. `tlmgr init-usertree`
+2. `kpsewhich -var-value=TEXMFHOME` should return
+   `~/texmf`
+3. `mkdir -p ~/texmf/tex/latex`
+4. `cd ~/texmf/tex/latex`
+5. `git clone
+   git@github.com:bvraghav/qptiet_latex-class tiet-question-paper`
+6. Verify that `kpsewhich tiet-question-paper.cls` finds `tiet-question-paper.cls`.
+
+## Usage ##
+
+### Document Class ###
+
+The document class is based on `article`.  And all the
+options are passed forward as is.
+
+```latex
+\documentclass[11pt,onecolumn]{tiet-question-paper}
+```
+
+### Title Block ###
+
+Only the following macros contribute to the title block.
+
+```latex
+\date{28 May 2024}
+% \institute{Thapar Institute of Engineering \&
+%   Technology}                   % Default value
+% \abbrevinst{\textsc{tiet}}      % Reserved for future
+% \instaddress{Patiala 147004}    % Reserved for future
+% \cohort{}                       % Reserved for future
+% \instlogo{images/tiet-logo.pdf} % Optional
+\schoolordepartment{Computer Science \& Engineering Department}
+\examname{End Semester Examination}
+\coursecode{UCS505}
+\coursename{Computer Graphics}
+\timeduration{3 hours}
+\maxmarks{45}
+\faculty{ANG,AMK,HPS,YDS,RGB}
+```
+
+### Geometry ###
+
+The class loads `geometry` package with following
+default options,
+
+```latex
+\RequirePackage[%
+  a4paper,%
+  left=1in,%
+  top=0.5in,%
+  right=0.75in,%
+  bottom=0.75in%
+]{geometry}
+```
+
+However, a user may reconfigure using `\geometry`
+command in [the
+preamble](https://duckduckgo.com/?q=what+is+a+preamble+in+latex+document)
+for example,
+
+```latex
+\geometry{letterpaper,bindingoffset=0.2in,
+          left=1.2in,right=1.2in,top=.8in,bottom=.8in,
+          footskip=.25in}
+```
+
+### Rules ###
+
+```latex
+\bvrhrule
+\bvrhrule[2pt]
+```
+Create a horizontal rule with length equal to line
+width and thickness `0.4pt` by default.  Optional
+argument specifies line thickness.  Here’s the
+implementation detail,
+
+```latex
+\DeclareRobustCommand%
+% Horizontal rule after each question
+\bvrhrule[1][0.4pt]{\rule{\linewidth}{#1}}
+```
+
+### Line Skip ###
+
+```latex
+\bvrskipline
+\bvrskipline[1.2]
+```
+
+Create vertical blank space with height specified as a
+factor of `\baselineskip`.  Optional argument specifies
+the factor, `1.0` by default.  Here’s the
+implementation detail,
+
+```latex
+\DeclareRobustCommand%
+% Skip line
+\bvrskipline[1][]{\vspace{#1\baselineskip}}
+```
+
+## Emacs Org Integration ##
+Add the following to [the
+init-file](https://www.gnu.org/software/emacs/manual/html_node/emacs/Init-File.html) 
+```elisp
+(add-to-list 'org-latex-classes
+	     `("tiet-question-paper" "\\documentclass{tiet-question-paper}
+[NO-DEFAULT-PACKAGES]
+\\usepackage{amsmath}
+\\usepackage{graphicx}
+\\usepackage{wrapfig}
+\\usepackage{amssymb}
+\\usepackage[unicode]{hyperref}
+" nil))
+```
+
+Here’s a [`sample-org`](./sample-org.org) file; that
+produces a [`PDF`](./sample-org.pdf)
diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/images/sample.png b/Master/texmf-dist/doc/latex/tiet-question-paper/images/sample.png
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diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.org b/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.org
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+#+date: \today
+
+#+latex_class: tiet-question-paper
+#+latex_class_options: [11pt]
+#+options: num:nil toc:nil author:nil email:nil 
+
+#+latex_header_extra: \hypersetup{%
+#+latex_header_extra:   colorlinks,%
+#+latex_header_extra:   breaklinks,%
+#+latex_header_extra:   urlcolor=[rgb]{0,0.35,0.65},%
+#+latex_header_extra:   linkcolor=[rgb]{0,0.35,0.65}%
+#+latex_header_extra: }
+
+#+latex_header_extra: \usepackage{libertinus}
+
+#+latex_header_extra: \instlogo{images/tiet-logo.pdf}
+#+latex_header_extra: \schoolordepartment{%
+#+latex_header_extra: Computer Science \& Engineering Department}
+#+latex_header_extra: \examname{End Semester Examination}
+#+latex_header_extra: \coursecode{UCS505}
+#+latex_header_extra: \coursename{Computer Graphics}
+#+latex_header_extra: \timeduration{3 hours}
+#+latex_header_extra: \maxmarks{45}
+#+latex_header_extra: \faculty{ANG,AMK,HPS,YDS,RGB}
+
+#+latex: \maketitle
+
+*Instructions:*
+1. Attempt any 5 questions;
+2. Attempt all the subparts of a question at one place.
+
+#+latex: \bvrhrule\bvrskipline
+
+1. 
+   1. Given the control polygon $\textbf{b}_0,
+      \textbf{b}_1, \textbf{b}_2, \textbf{b}_3$ of a
+      Cubic Bezier curve; determine the vertex
+      coordinates for parameter values $\forall t\in
+      T$. \hfill [7 marks]
+      \begin{align*}
+        T \equiv
+        & \{0, 0.15, 0.35, 0.5, 0.65, 0.85, 1\} \\
+        \begin{bmatrix}
+          \textbf{b}_0 &\textbf{b}_1& \textbf{b}_2& \textbf{b}_3
+        \end{bmatrix} \equiv& \begin{bmatrix}
+          1&2&4&3\\ 1&3&3&1
+        \end{bmatrix}
+      \end{align*}
+
+   2. Explain the role of convex hull in curves.
+      \hfill[2 marks]
+
+#+latex: \bvrhrule
+
+#+ATTR_LATEX: :options [resume]
+1. 
+   1. Describe the continuity conditions for
+      curvilinear geometry.  \hfill[5 marks]
+   2. Define formally, a B-Spline curve. \hfill [2
+      marks]
+   3. How is a Bezier curve different from a B-Spline
+      curve? \hfill [2 marks]
+
+#+latex: \bvrhrule
+
+#+ATTR_LATEX: :options [resume]
+1. 
+   1. Given a triangle, with vertices defined by column
+      vectors of $P$; find its vertices after
+      reflection across XZ plane. \hfill [3 marks]
+      \begin{align*}
+        P\equiv
+        &\begin{bmatrix}
+          3&6&5 \\ 4&4&6 \\ 1&2&3
+        \end{bmatrix}
+      \end{align*}
+   2. Given a pyramid with vertices defined by the
+      column vectors of $P$, and an axis of rotation
+      $A$ with direction $\textbf{v}$ and passing
+      through $\textbf{p}$.  Find the coordinates of
+      the vertices after rotation about $A$ by an angle
+      of $\theta=\pi/4$.\hfill [6 marks]
+      \begin{align*}
+        P\equiv
+        &\begin{bmatrix}
+          0&1&0&0 \\ 0&0&1&0 \\0&0&0&1
+        \end{bmatrix} \\
+        \begin{bmatrix}
+          \mathbf{v} & \mathbf{p}
+        \end{bmatrix}\equiv
+        &\begin{bmatrix}
+          0&0 \\1&1\\1&0
+        \end{bmatrix}
+      \end{align*}
+#+latex: \bvrhrule
+
+#+ATTR_LATEX: :options [resume]
+1. 
+   1. Explain the two winding number rules for inside
+      outside tests. \hfill [4 marks]
+   2. Explain the working principle of a CRT. \hfill [5
+      marks]
+
+#+latex: \bvrhrule
+
+#+ATTR_LATEX: :options [resume]
+1. 
+   1. Given a projection plane $P$ defined by normal
+      $\textbf{n}$ and a reference point $\textbf{a}$;
+      and the centre of projection as $\mathbf{p}_0$;
+      find the perspective projection of the point
+      $\textbf{x}$ on $P$. \hfill [5 marks]
+      \begin{align*}
+        \begin{bmatrix}
+          \mathbf{a}&\mathbf{n}&\mathbf{p}_0&\mathbf{x}
+        \end{bmatrix}\equiv
+        &
+          \begin{bmatrix}
+            3&-1&1&8\\4&2&1&10\\5&-1&3&6
+          \end{bmatrix}
+      \end{align*}
+   2. Given a geometry $G$, which is a standard unit
+      cube scaled uniformly by half and viewed through
+      a Cavelier projection bearing $\theta=\pi/4$
+      wrt. $X$ axis. \hfill [2 marks]
+   3. Given a view coordinate system (VCS) with origin
+      at $\textbf{p}_v$ and euler angles ZYX as
+      $\boldsymbol{\theta}$ wrt. the world coordinate
+      system (WCS); find the location $\mathbf{x}_v$ in
+      VCS, corresponding to $\textbf{x}_w$ in
+      WCS. \hfill [2 marks]
+      \begin{align*}
+        \begin{bmatrix}
+          \mathbf{p}_v & \boldsymbol{\theta} & \mathbf{x}_w
+        \end{bmatrix}\equiv
+        &\begin{bmatrix}
+          5&\pi/3&10\\5&0&10\\0&0&0
+        \end{bmatrix}
+      \end{align*}
+
+#+latex: \bvrhrule
+
+#+ATTR_LATEX: :options [resume]
+1. 
+   1. Describe the visible surface detection problem in
+      about 25 words. \hfill [1 mark]
+   2. To render a scene with $N$ polygons into a
+      display with height $H$; what are the space and
+      time complexities respectively of a typical
+      image-space method. \hfill [2 marks]
+   3. Given a 3D space bounded within $[0\quad0\quad0]$
+      and $[7\quad7\quad-7]$, containing two infinite
+      planes each defined by 3 incident points
+      $\mathbf{a}_0, \mathbf{a}_1, \mathbf{a}_2$ and
+      $\mathbf{b}_0, \mathbf{b}_1, \mathbf{b}_2$
+      respectively bearing colours (RGB) as
+      $\mathbf{c}_a$ and $\textbf{c}_b$ respectively.
+      \begin{align*}
+        \begin{bmatrix}
+          \mathbf{a}_0&\mathbf{a}_1&\mathbf{a}_2
+          &\mathbf{b}_0&\mathbf{b}_1&\mathbf{b}_2
+          &\mathbf{c}_a&\mathbf{c}_b
+        \end{bmatrix}\equiv
+        &\begin{bmatrix}
+          1&6&1&6&1&6&1&0 \\
+          1&3&6&6&3&1&0&0 \\
+          -1&-6&-1&-1&-6&-1&0&1
+        \end{bmatrix}
+      \end{align*}
+      Compute and/ or determine using the depth-buffer
+      method, the colour at pixel $\mathbf{x}=(2,4)$ on
+      a display resolved into $7\times7$ pixels. The
+      projection plane is at $Z=0$, looking at
+      $-Z$. \hfill [6 marks]
+
+#+latex: \bvrhrule
+
+
+# Local Variables:
+# org-latex-default-packages-alist: nil
+# org-latex-packages-alist: nil
+# End:
diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.pdf b/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.pdf
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diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.tex b/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample-org.tex
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+% Created 2024-06-22 Sat 22:57
+% Intended LaTeX compiler: pdflatex
+\documentclass[11pt]{tiet-question-paper}
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{amssymb}
+\usepackage[unicode]{hyperref}
+
+
+\hypersetup{%
+colorlinks,%
+breaklinks,%
+urlcolor=[rgb]{0,0.35,0.65},%
+linkcolor=[rgb]{0,0.35,0.65}%
+}
+\usepackage{libertinus}
+\instlogo{images/tiet-logo.pdf}
+\schoolordepartment{%
+Computer Science \& Engineering Department}
+\examname{End Semester Examination}
+\coursecode{UCS505}
+\coursename{Computer Graphics}
+\timeduration{3 hours}
+\maxmarks{45}
+\faculty{ANG,AMK,HPS,YDS,RGB}
+\date{\today}
+\title{}
+\hypersetup{
+ pdfauthor={B.V. Raghav},
+ pdftitle={},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 29.3 (Org mode 9.6.15)}, 
+ pdflang={English}}
+\begin{document}
+
+\maketitle
+
+\textbf{Instructions:}
+\begin{enumerate}
+\item Attempt any 5 questions;
+\item Attempt all the subparts of a question at one place.
+\end{enumerate}
+
+\bvrhrule\bvrskipline
+
+\begin{enumerate}
+\item \begin{enumerate}
+\item Given the control polygon \(\textbf{b}_0,
+      \textbf{b}_1, \textbf{b}_2, \textbf{b}_3\) of a
+Cubic Bezier curve; determine the vertex
+coordinates for parameter values \(\forall t\in
+      T\). \hfill [7 marks]
+\begin{align*}
+  T \equiv
+  & \{0, 0.15, 0.35, 0.5, 0.65, 0.85, 1\} \\
+  \begin{bmatrix}
+    \textbf{b}_0 &\textbf{b}_1& \textbf{b}_2& \textbf{b}_3
+  \end{bmatrix} \equiv& \begin{bmatrix}
+    1&2&4&3\\ 1&3&3&1
+  \end{bmatrix}
+\end{align*}
+
+\item Explain the role of convex hull in curves.
+\hfill[2 marks]
+\end{enumerate}
+\end{enumerate}
+
+\bvrhrule
+
+\begin{enumerate}[resume]
+\item \begin{enumerate}
+\item Describe the continuity conditions for
+curvilinear geometry.  \hfill[5 marks]
+\item Define formally, a B-Spline curve. \hfill [2
+marks]
+\item How is a Bezier curve different from a B-Spline
+curve? \hfill [2 marks]
+\end{enumerate}
+\end{enumerate}
+
+\bvrhrule
+
+\begin{enumerate}[resume]
+\item \begin{enumerate}
+\item Given a triangle, with vertices defined by column
+vectors of \(P\); find its vertices after
+reflection across XZ plane. \hfill [3 marks]
+\begin{align*}
+  P\equiv
+  &\begin{bmatrix}
+    3&6&5 \\ 4&4&6 \\ 1&2&3
+  \end{bmatrix}
+\end{align*}
+\item Given a pyramid with vertices defined by the
+column vectors of \(P\), and an axis of rotation
+\(A\) with direction \(\textbf{v}\) and passing
+through \(\textbf{p}\).  Find the coordinates of
+the vertices after rotation about \(A\) by an angle
+of \(\theta=\pi/4\).\hfill [6 marks]
+\begin{align*}
+  P\equiv
+  &\begin{bmatrix}
+    0&1&0&0 \\ 0&0&1&0 \\0&0&0&1
+  \end{bmatrix} \\
+  \begin{bmatrix}
+    \mathbf{v} & \mathbf{p}
+  \end{bmatrix}\equiv
+  &\begin{bmatrix}
+    0&0 \\1&1\\1&0
+  \end{bmatrix}
+\end{align*}
+\end{enumerate}
+\end{enumerate}
+\bvrhrule
+
+\begin{enumerate}[resume]
+\item \begin{enumerate}
+\item Explain the two winding number rules for inside
+outside tests. \hfill [4 marks]
+\item Explain the working principle of a CRT. \hfill [5
+marks]
+\end{enumerate}
+\end{enumerate}
+
+\bvrhrule
+
+\begin{enumerate}[resume]
+\item \begin{enumerate}
+\item Given a projection plane \(P\) defined by normal
+\(\textbf{n}\) and a reference point \(\textbf{a}\);
+and the centre of projection as \(\mathbf{p}_0\);
+find the perspective projection of the point
+\(\textbf{x}\) on \(P\). \hfill [5 marks]
+\begin{align*}
+  \begin{bmatrix}
+    \mathbf{a}&\mathbf{n}&\mathbf{p}_0&\mathbf{x}
+  \end{bmatrix}\equiv
+  &
+    \begin{bmatrix}
+      3&-1&1&8\\4&2&1&10\\5&-1&3&6
+    \end{bmatrix}
+\end{align*}
+\item Given a geometry \(G\), which is a standard unit
+cube scaled uniformly by half and viewed through
+a Cavelier projection bearing \(\theta=\pi/4\)
+wrt. \(X\) axis. \hfill [2 marks]
+\item Given a view coordinate system (VCS) with origin
+at \(\textbf{p}_v\) and euler angles ZYX as
+\(\boldsymbol{\theta}\) wrt. the world coordinate
+system (WCS); find the location \(\mathbf{x}_v\) in
+VCS, corresponding to \(\textbf{x}_w\) in
+WCS. \hfill [2 marks]
+\begin{align*}
+  \begin{bmatrix}
+    \mathbf{p}_v & \boldsymbol{\theta} & \mathbf{x}_w
+  \end{bmatrix}\equiv
+  &\begin{bmatrix}
+    5&\pi/3&10\\5&0&10\\0&0&0
+  \end{bmatrix}
+\end{align*}
+\end{enumerate}
+\end{enumerate}
+
+\bvrhrule
+
+\begin{enumerate}[resume]
+\item \begin{enumerate}
+\item Describe the visible surface detection problem in
+about 25 words. \hfill [1 mark]
+\item To render a scene with \(N\) polygons into a
+display with height \(H\); what are the space and
+time complexities respectively of a typical
+image-space method. \hfill [2 marks]
+\item Given a 3D space bounded within \([0\quad0\quad0]\)
+and \([7\quad7\quad-7]\), containing two infinite
+planes each defined by 3 incident points
+\(\mathbf{a}_0, \mathbf{a}_1, \mathbf{a}_2\) and
+\(\mathbf{b}_0, \mathbf{b}_1, \mathbf{b}_2\)
+respectively bearing colours (RGB) as
+\(\mathbf{c}_a\) and \(\textbf{c}_b\) respectively.
+\begin{align*}
+  \begin{bmatrix}
+    \mathbf{a}_0&\mathbf{a}_1&\mathbf{a}_2
+    &\mathbf{b}_0&\mathbf{b}_1&\mathbf{b}_2
+    &\mathbf{c}_a&\mathbf{c}_b
+  \end{bmatrix}\equiv
+  &\begin{bmatrix}
+    1&6&1&6&1&6&1&0 \\
+    1&3&6&6&3&1&0&0 \\
+    -1&-6&-1&-1&-6&-1&0&1
+  \end{bmatrix}
+\end{align*}
+Compute and/ or determine using the depth-buffer
+method, the colour at pixel \(\mathbf{x}=(2,4)\) on
+a display resolved into \(7\times7\) pixels. The
+projection plane is at \(Z=0\), looking at
+\(-Z\). \hfill [6 marks]
+\end{enumerate}
+\end{enumerate}
+
+\bvrhrule
+\end{document}
diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample.pdf b/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample.pdf
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diff --git a/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample.tex b/Master/texmf-dist/doc/latex/tiet-question-paper/tiet-question-paper-sample.tex
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+\documentclass[11pt,a4paper,onecolumn]{tiet-question-paper}
+\date{28 May 2024}
+\institute{Alpha}
+\instlogo{images/tiet-logo.pdf}
+\schoolordepartment{%
+  Computer Science \& Engineering Department}
+\examname{%
+  End Semester Examination}
+\coursecode{UCS505}
+\coursename{Computer Graphics}
+\timeduration{3 hours}
+\maxmarks{45}
+\faculty{ANG,AMK,HPS,YDS,RGB}
+
+\begin{document}
+\maketitle
+\textbf{Instructions:}
+\begin{enumerate}
+\item Attempt any 5 questions;
+\item Attempt all the subparts of a question at one
+  place.
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}
+\item
+  \begin{enumerate}
+  \item Given the control polygon
+    $\textbf{b}_0, \textbf{b}_1, \textbf{b}_2,
+    \textbf{b}_3$ of a Cubic Bezier curve; determine
+    the coordinates for parameter values
+    $\forall t\in T$. \hfill [7 marks]
+
+    \begin{align*}
+      T \equiv
+      & \{0, 0.15, 0.35, 0.5, 0.65, 0.85, 1\} \\
+      \begin{bmatrix}
+        \textbf{b}_0 &\textbf{b}_1& \textbf{b}_2& \textbf{b}_3
+      \end{bmatrix} \equiv
+      & \begin{bmatrix}
+        1&2&4&3\\ 1&3&3&1
+      \end{bmatrix}
+    \end{align*}
+  \item Explain the role of convex hull in curves.
+    \hfill[2 marks]
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}[resume]
+\item
+  \begin{enumerate}
+  \item Describe the continuity conditions for
+    curvilinear geometry.
+    \hfill[5 marks]
+  \item Define formally, a B-Spline curve. \hfill [2
+    marks]
+  \item How is a Bezier curve different from a B-Spline
+    curve?
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}[resume]
+\item
+  \begin{enumerate}
+  \item Given a triangle, with vertices defined by
+    column vectors of $P$; find its vertices after
+    reflection across XZ plane. \hfill [3 marks]
+    \begin{align*}
+      P\equiv
+      &\begin{bmatrix}
+        3&6&5 \\ 4&4&6 \\ 1&2&3
+      \end{bmatrix}
+    \end{align*}
+  \item Given a pyramid with vertices defined by the
+    column vectors of $P$, and an axis of rotation $A$
+    with direction $\textbf{v}$ and passing through
+    $\textbf{p}$.  Find the coordinates of the vertices
+    after rotation about $A$ by an angle of
+    $\theta=\pi/4$.\hfill [6 marks]
+    \begin{align*}
+      P\equiv
+      &\begin{bmatrix}
+        0&1&0&0 \\ 0&0&1&0 \\0&0&0&1
+      \end{bmatrix} \\
+      \begin{bmatrix}
+        \mathbf{v} & \mathbf{p}
+      \end{bmatrix}\equiv
+      &\begin{bmatrix}
+        0&0 \\1&1\\1&0
+      \end{bmatrix}
+    \end{align*}
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}[resume]
+\item
+  \begin{enumerate}
+    \item Explain the two winding number rules for
+      inside outside tests. \hfill [4 marks]
+    \item Explain the working principle of a
+      CRT. \hfill [5 marks]
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}[resume]
+\item
+  \begin{enumerate}
+  \item Given a projection plane $P$ defined by normal
+    $\textbf{n}$ and a reference point $\textbf{a}$;
+    and the centre of projection as $\mathbf{p}_0$;
+    find the perspective projection of the point
+    $\textbf{x}$ on $P$. \hfill [5 marks]
+    \begin{align*}
+      \begin{bmatrix}
+        \mathbf{a}&\mathbf{n}&\mathbf{p}_0&\mathbf{x}
+      \end{bmatrix}\equiv
+      &
+        \begin{bmatrix}
+          3&-1&1&8\\4&2&1&10\\5&-1&3&6
+        \end{bmatrix}
+    \end{align*}
+  \item Given a geometry $G$, which is a standard unit
+    cube scaled uniformly by half and viewed through a
+    Cavelier projection bearing $\theta=\pi/4$
+    wrt. $X$-axis. \hfill [2 marks]
+  \item Given a view coordinate system (VCS) with
+    origin at $\textbf{p}_v$ and euler angles ZYX
+    $\boldsymbol{\theta}$ wrt. world coordinate system
+    (WCS); find the location $\mathbf{x}_v$ in VCS,
+    corresponding to the point $\textbf{x}_w$ in
+    WCS. \hfill [2 marks]
+    \begin{align*}
+      \begin{bmatrix}
+        \mathbf{p}_v & \boldsymbol{\theta} & \mathbf{x}_w
+      \end{bmatrix}\equiv
+      &\begin{bmatrix}
+        5&\pi/3&10\\5&0&10\\0&0&0
+      \end{bmatrix}
+    \end{align*}
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\begin{enumerate}[resume]
+\item
+  \begin{enumerate}
+    \item Describe the visible surface detection
+      problem in about 25 words. \hfill [1 mark]
+    \item To render a scene with $N$ polygons into a
+      display with height $H$; what are the space and
+      time complexities respectively of a typical
+      image-space method. \hfill [2 marks]
+    \item Given a 3D space bounded within
+      $[0\quad0\quad0]$ and $[7\quad7\quad-7]$,
+      containing two infinite planes each defined by 3
+      incident points
+      $\mathbf{a}_0, \mathbf{a}_1, \mathbf{a}_2$ and
+      $\mathbf{b}_0, \mathbf{b}_1, \mathbf{b}_2$
+      respectively bearing colours (RGB) as
+      $\mathbf{c}_a$ and $\textbf{c}_b$ respectively.
+      \begin{align*}
+        \begin{bmatrix}
+          \mathbf{a}_0&\mathbf{a}_1&\mathbf{a}_2
+          &\mathbf{b}_0&\mathbf{b}_1&\mathbf{b}_2
+          &\mathbf{c}_a&\mathbf{c}_b
+        \end{bmatrix}\equiv
+        &\begin{bmatrix}
+          1&6&1&6&1&6&1&0 \\
+          1&3&6&6&3&1&0&0 \\
+          -1&-6&-1&-1&-6&-1&0&1
+        \end{bmatrix}
+      \end{align*}
+
+      Compute and/ or determine using the depth-buffer
+      method, the colour at pixel $\mathbf{x}=(2,4)$ on
+      a display resolved into $7\times7$ pixels. The
+      projection plane is at $Z=0$, looking at
+      $-Z$. \hfill [6 marks]
+  \end{enumerate}
+\end{enumerate}
+\bvrhrule[0.4pt]
+\end{document}