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\documentclass[11pt]{tiet-question-paper}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{amssymb}
\usepackage[unicode]{hyperref}


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colorlinks,%
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\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}
