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exam2.pdf

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PANDOC := pandoc
exam2.pdf: exam2.md
$(PANDOC) -o $@ $<

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---
geometry: margin=2cm
output: pdf_document
title: Exam 2
subtitle: CSCI 5607
date: \today
author: |
| Michael Zhang
| zhan4854@umn.edu $\cdot$ ID: 5289259
---
\renewcommand{\c}[1]{\textcolor{gray}{#1}}
## Reflection and Refraction
1. Consider a sphere $S$ made of solid glass ($\eta$ = 1.5) that has radius $r =
3$ and is centered at the location $s = (2, 2, 10)$ in a vaccum ($\eta =
1.0$). If a ray emanating from the point $e = (0, 0, 0)$ intersects $S$ at a
point $p = (1, 4, 8)$:
a. (2 points) What is the angle of incidence $\theta_i$ ?
First, the normal at the point $(1, 4, 8)$ is determined by subtracting
that point from the center $(2, 2, 10)$, which gets us $N = (2 - 1, 2 - 4,
10 - 8) = (1, -2, 2)$. Then, to determine the angle between
b. (1 points) What is the angle of reflection $\theta_r$ ?
c. (3 points) What is the direction of the reflected ray?
d. (3 points) What is the angle of transmission $\theta_t$ ?
e. (4 points) What is the direction of the transmitted ray?
Using Snell's law, we know that $\eta_1 \sin \theta_1 = \eta_2 \sin
\theta_2$. In this case, let material 1 be the vacuum, and material 2 be
the glass. Then, we have $1.0 \times \sin \theta_1$
## Geometric Transformations
2. \c{(8 points) Consider the airplane model below, defined in object
coordinates with its center at $(0, 0, 0)$, its wings aligned with the $\pm
x$ axis, its tail pointing upwards in the $+y$ direction and its nose facing
in the $+z$ direction. Derive a sequence of model transformation matrices
that can be applied to the vertices of the airplane to position it in space
at the location $p = (4, 4, 7)$, with a direction of flight $w = (2, 1, 2)$
and the wings aligned with the direction $d = (2, 2, 1)$.}
The translation matrix is
$$
\begin{bmatrix}
1 & 0 & 0 & x \\
0 & 1 & 0 & y \\
0 & 0 & 1 & z \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & 4 \\
0 & 1 & 0 & 4 \\
0 & 0 & 1 & 7 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
$$
Since the direction of flight was originally $(0, 0, 1)$, we have to
transform it to $(2, 1, -2)$.
## Clipping
9. \c{Consider the triangle whose vertex positions, after the viewport
transformation, lie in the centers of the pixels: $p_0 = (3, 3), p_1 = (9,
5), p_2 = (11, 11)$.}
Starting at $p_0$, the three vectors are:
- $v_0 = p_1 - p_0 = (9 - 3, 5 - 3) = (6, 2)$
- $v_1 = p_2 - p_1 = (11 - 9, 11 - 5) = (2, 6)$
- $v_2 = p_0 - p_2 = (3 - 11, 3 - 11) = (-8, -8)$
The first edge vector $e$ would be $(6, 2)$, and the edge normal would be
that rotated by $90^\circ$.
a. \c{(6 points) Define the edge equations and tests that would be applied,
during the rasterization process, to each pixel $(x, y)$ within the bounding
rectangle $3 \le x \le 11, 3 \le y \le 11$ to determine if that pixel is
inside the triangle or not.}
b. \c{(3 points) Consider the three pixels $p_4 = (6, 4), p_5 = (7, 7)$, and
$p_6 = (10, 8)$. Which of these would be considered to lie inside the
triangle, according to the methods taught in class?}

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texlive.combined.scheme-full texlive.combined.scheme-full
unzip unzip
zip zip
zathura
(python310.withPackages (p: with p; [ ipython numpy scipy sympy ])) (python310.withPackages (p: with p; [ ipython numpy scipy sympy ]))
]) ++ (with toolchain; [ ]) ++ (with toolchain; [