csci5607/exam-2/exam2.md

geometry	output	title	subtitle	date	author
margin=2cm	pdf_document	Exam 2	CSCI 5607	\today	| Michael Zhang | zhan4854@umn.edu $\cdot$ ID: 5289259

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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?}