Exam 2

exam-2/.gitignore

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+exam2.pdf

exam-2/Makefile

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

exam-2/exam2.md

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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?}
+
+

flake.nix

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