\begin
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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}