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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}