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#+date: \today
#+latex_class: tiet-question-paper
#+latex_class_options: [11pt]
#+options: num:nil toc:nil author:nil email:nil
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#+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
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