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+% 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}
+
+
+\hypersetup{%
+colorlinks,%
+breaklinks,%
+urlcolor=[rgb]{0,0.35,0.65},%
+linkcolor=[rgb]{0,0.35,0.65}%
+}
+\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}