<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% 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}


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colorlinks,%
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\usepackage{libertinus}
\instlogo{images/tiet-logo.pdf}
\schoolordepartment{%
Computer Science \&amp; 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
  &amp; \{0, 0.15, 0.35, 0.5, 0.65, 0.85, 1\} \\
  \begin{bmatrix}
    \textbf{b}_0 &amp;\textbf{b}_1&amp; \textbf{b}_2&amp; \textbf{b}_3
  \end{bmatrix} \equiv&amp; \begin{bmatrix}
    1&amp;2&amp;4&amp;3\\ 1&amp;3&amp;3&amp;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
  &amp;\begin{bmatrix}
    3&amp;6&amp;5 \\ 4&amp;4&amp;6 \\ 1&amp;2&amp;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
  &amp;\begin{bmatrix}
    0&amp;1&amp;0&amp;0 \\ 0&amp;0&amp;1&amp;0 \\0&amp;0&amp;0&amp;1
  \end{bmatrix} \\
  \begin{bmatrix}
    \mathbf{v} &amp; \mathbf{p}
  \end{bmatrix}\equiv
  &amp;\begin{bmatrix}
    0&amp;0 \\1&amp;1\\1&amp;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}&amp;\mathbf{n}&amp;\mathbf{p}_0&amp;\mathbf{x}
  \end{bmatrix}\equiv
  &amp;
    \begin{bmatrix}
      3&amp;-1&amp;1&amp;8\\4&amp;2&amp;1&amp;10\\5&amp;-1&amp;3&amp;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 &amp; \boldsymbol{\theta} &amp; \mathbf{x}_w
  \end{bmatrix}\equiv
  &amp;\begin{bmatrix}
    5&amp;\pi/3&amp;10\\5&amp;0&amp;10\\0&amp;0&amp;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&amp;\mathbf{a}_1&amp;\mathbf{a}_2
    &amp;\mathbf{b}_0&amp;\mathbf{b}_1&amp;\mathbf{b}_2
    &amp;\mathbf{c}_a&amp;\mathbf{c}_b
  \end{bmatrix}\equiv
  &amp;\begin{bmatrix}
    1&amp;6&amp;1&amp;6&amp;1&amp;6&amp;1&amp;0 \\
    1&amp;3&amp;6&amp;6&amp;3&amp;1&amp;0&amp;0 \\
    -1&amp;-6&amp;-1&amp;-1&amp;-6&amp;-1&amp;0&amp;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}
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