Exams

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Computer Graphics - $2022/2023$

1. (1 point) We want to fly a model helicopter around a target placed at the origin $(0,0,0)$ .

The helicopter should fly at a constant altitude of 5 units following a circle of radius of (also) 5 units and be tilted 45 degrees towards the ground. The helicopter rotates at a speed of 15 degrees per second in the counterclockwise direction. At time $t = 0$ the helicopter is at position $C_0 = (0.0, 5.0, 5.0)$ .

The image below illustrates the situation. On the left, a side view of the helicopter at position $C_0$ . On the right, a top view of the helicopter at position $C_0$ and at position $C_1$ after rotating degrees. The red point is the origin.

Which of the following matrices represents the helicopter's orientation and position in regard to the origin after $3$ seconds?

Consider the initial position of the helicopter as centred at the origin and facing the $-z$ axis.

Image /page/0/Figure/6 description: The image contains two diagrams illustrating a movement or trajectory. The first diagram shows a starting point labeled "C0" with an arrow indicating a direction at a 45-degree angle. There are dashed lines indicating a distance of "5" units both vertically and horizontally from the starting point to a red dot. The second diagram shows a circular path with a radius of "5" units. The starting point "C0" is at the top of the circle, and an arrow indicates a direction tangent to the circle. Another point is shown on the circle, and an angle "θ" is marked between the z-axis and the line connecting the center of the circle to this point. The x, y, and z axes are also labeled in both diagrams.

Image /page/0/Figure/7 description: The image shows eight matrices labeled A through H. Each matrix is a 4x4 matrix containing a mix of numbers, including fractions, square roots, and integers. The matrices are arranged in two rows of four, with the labels positioned to the left of each matrix.

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1. (1 point) We want to take a photo from inside a house looking out a window using a pinhole camera. The camera is mounted on a tripod that maintains its view vector in the same orientation as the x-axis. The coordinates of the window's corners are $(5.0, 0.8, -2.0), (5.0, 0.8, -3.6), (5.0, 3.8, -2.0), (5.0, 3.8, -3.6),$ and the origin is inside the house. Given that the camera has fixed focal length and aspect ratio equals 1, where would you place the camera so that it covers the entire window and, at the same time, only captures the minimum amount of wall (or anything but the window)?

A. $(5.0, 2.8, -2.8)$ B. $(3.0, 2.3, -3.6)$ C. $(0.5, 0.8, -3.6)$ D. E. F. G. H.

1. $(1 \text{ point})$ A square in 3D space is projected to the screen using a linear perspective transformation. The 2D coordinates of the square after projection are: $(-1.0, -1.0), (1.0, -1.0), (-0.6, 1.0),$ and Considering the vanishing lines generated by the projected square, what are the 2D coordinates of the vanishing point that is not at infinity?

A. $(0.0, 6.5)$ B. $(0.5, 3.5)$ C. $(1.0, 5.0)$ D. $(0.0, 3.75)$ E. F. G. H.

1. (1 point) We have two point lights with RGB emission $I_1 = (1.0, G, 0.6)$ and $I_2 = (0.5, 1.0, B)$ , where the values G and B can be chosen in the range [0,1]. The lights are located at positions and .

Consider the floor as the plane $y = 0$ , normal $n = (0, 1, 0)$ and made of a completely diffuse material with coefficient $k_d = (0.8, 0.8, 1.0)$ .

To which values should we set the light parameters $G$ and $B$ in order for an observer to perceive the point $p = (0, 0, 0)$ on the floor as a shade of gray?

Obs: Think how shades of gray are represented in RGB.

A. $G = 0.2; B = 0.8$ B. $G = 0.5; B = 0.5$ C. $G = 0.7; B = 0.2$ D. E. F. G. H.

1. (1 point) We want to install a projector. Unfortunately, it cannot be placed perfectly aligned with the diffuse screen. Assuming the projection direction along the optical axis of the projector and the screen normal form an angle of $\frac{3}{4}\pi$ (radians), how much would one need to scale up the brightness of the projector, in order to deliver the same projected image brightness in the projection centre as a perfectly

B. $2\pi$ C. $4/\pi$ D. 1 E. $\sqrt{2}$ F. 1.75 G. $$ H. 2 A.

$6. (1 point)$ LED lamps often combine several colored lamps to simulate white light. Assume $S1, S2, S3$ are light sources with RGB emission $\alpha_1$ (0.3,0.4,0.2), (0.2,0.1,0.2), (0.2,0.1,0.4) respectively. Calculate the values of , such that the added emission matches .

What is the result of $\alpha_1^2 + 2.5\alpha_2 + \alpha_3/2?$

aligned projector at the same distance?

A. 5 B. 6 C. 7 D. 8 E. 9

F. 9.2 G. 9.5 H. 10 I. 11.4

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1. (1 point) Consider a point light source at position $(2\sqrt{3}, 2, 0)$ on the ceiling of a room (plane y=2), and a reflective sphere of radius $r = \sqrt{2}$ centered at . Further, consider the sphere material as a mirror.

Given the point $s = (0, 0, 0)$ on the sphere, at which point will the light be reflected on the ceiling?

A. $\left(\frac{2}{\sqrt{2}}, \sqrt{2}, 0\right)$ B. C. D. E. F. G. H.

1. (1 point) During triangle rasterization you can use an Axis-Aligned-Bounding-Box (AABB) to reduce the search space; limit the number of pixels whose centers will be checked for being inside/outside the triangle.

Consider now that we want to approximate $2D$ circles by using a $GL\_TRIANGLE\_FAN$ primitive. It is similar to $GL\_TRIANGLE\_STRIP$ , but after the three first vertices, every new vertex connects to the first and last vertex. So, for a sequence of vertices

$v_0, v_1, v_2, v_3, v_4, v_5$ the following triangles are created

$v_0, v_1, v_2,$ ${}_{}$

v_0, v_4, v_5.

}

To create approximations of a circle in 2D space, we use the method below. Note that we are using 2D vertices instead of $3D$ .

// input is center of circle and vector from center to first point
void \ draw \ (c_{-}x \ , \ c_{-}y \ , \ v_{-}x \ , \ v_{-}y) \ \{

float x = v_x;float \ y = v_{-y};float \cos = 1.0 \text{ f/sqrt} (2.0 \text{ f});float \sin = 1.0 \text{ f/sqrt} (2.0 \text{ f});\text{glBegin}(\text{GL\_TRIANGLE\_FAN});glVertex2f (c_x, c_y);\text{glVertex2f} \quad (\quad \text{c\_x+x} \quad \text{c\_y+y} \;);for (int i = 0; i < 8; +\!\!+\!\!i) {
      \text{float px} = \cos*x - \sin*y;\text{float } \text{py} = \cos \ast x + \sin \ast y;\text{glVertex2f (}c_x+px, c_y+py);x = px;y = py;\text{glEnd}();

The rasterizer is now modified to create and then rely on AABBs for these fans that approximate circles. If we call the method with parameters

draw $(3.5, 4.3, 2.0/\sqrt{2.0}), 2.0/\sqrt{2.0})$

what are the coordinates of the four corners of the respective AABB?

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1. (1 point) The following code draws an image by setting the pixel colors. Which of the following images corresponds to the output of this code?

(the $mod(x,y)$ function returns $x \mod y$ )

int x = mod(coordx, 10);int y = mod(coordy, 8);if (y = 0 \mid | y = 4)return false;
  \textbf{if} \ (\text{y} < 4 \ \&\& \ \text{x} == 4)return false;
  \textbf{if} \ (y > 4 \ \&\& \ x == 9)return false;
  return true;
}
<pre>void CreateImage () {</pre>
  for (int i=0; i < 20; ++i)
    for (int j=0; j<20; \pmj)
    {
       if (PixelColor(i,j))
         Img.pixel(i,j) = (1.0, 0.0, 0.0);
       elseImg.pixel(i,j) = (0.2, 0.2, 0.2);
    }

Image /page/3/Figure/5 description: The image shows eight different brick patterns, labeled A through H. The patterns are all red bricks with black mortar. Pattern A has vertical bricks with horizontal mortar lines. Pattern B has a horizontal band of mortar in the middle. Pattern C has a standard brick pattern. Pattern D has a more random brick pattern. Pattern E has a standard brick pattern. Pattern F has a horizontal band of mortar in the middle. Pattern G has a grid pattern. Pattern H has a more random brick pattern.

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1. (1 point) A two-step texture coordinates lookup uses an intermediate texture to look up the final texture coordinates values. In this case, given initial coordinates $u_i v_i$ the intermediate texture is bilinearly interpolated to lookup the texture coordinates of the final texture $u_f v_f$ . Consider the intermediate texture as defined below, where each texel holds a pair of numbers (in blue) representing the final

As an example, for initial texture coordinates $u_i v_i = \left(\frac{1}{3}, \frac{1}{5}\right)$ the final texture coordinates would be .

What would then be the final texture coordinates $u_f v_f$ when accessing initial texture coordinates $u_i v_i =$

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1. (1 point) We want to display a button on an interface. The button is represented with a texture that requires X bytes of memory and, depending on the device, we know that it will be stretched or squeezed along its x-axis only. To address aliasing on different devices, we apply a 1D mipmapping only along the x-axis (two horizontally neighboring texels are merged).

How much memory do we need to store all mipmap levels (including the original texture), if the original button texture has a width of $2^n$ and a height of $2^{2n}$ .

The solution can be obtained in several ways, similar to the solution for original Mipmaps.

• A. $2n^2X$ B. $\frac{2^{n+1}-1}{2^n}X$ C. D. E. F. G. H.

The sketch is not up to scale.

A. 1.5 B. 2 C. 3 D. 4 E. 4.5 F. 6 G. 7.5 $\text{H. 9}$

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1. (1 point) Using the Phong shading model (and the notation from class), we want to simulate the light interaction with a pink shiny plastic ball in a green room under a light cyan illumination of RGB value $Id=Is=(0.2,1,1)$ . Assume a shininess of 1, what are the values Ks, Kd, Ia (we assume Ka=Kd) that you would choose, if you want to opt for a realistic appearance? Please estimate suitable RGB values and explain your choices briefly (one sentence each).

For reference: a green room under white light and a pink plastic pink ball under white light.

Image /page/6/Picture/2 description: The image shows an empty room with green walls, floor, and ceiling. The room is rectangular and appears to be illuminated by a soft, diffused light. There are no objects or furniture in the room, and the walls are smooth and featureless. The perspective is from a slightly elevated viewpoint, looking down into the room.

Image /page/6/Picture/3 description: The image shows a close-up of several pink plastic balls. The balls are all the same size and color, and they are arranged in a random pattern. The balls are smooth and shiny, and they reflect light. The image is well-lit and in focus.

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Computer Graphics - $2020/2021$ Resit Examination

1. Given a $512 \times 512$ grayscale image, if you apply a $3 \times 3$ Box filter (averages nine pixels) will the result be the same as applying first a $1 \times 3$ average filter followed by a $3 \times 1$ average filter? Both smaller filters compute the average of three pixels. Explain your answer (only answering yes/no does not lead to any points). You can also ignore any border treatment for your explanation.

You can use the following $3 \times 3$ image patch to help in your explanation. For example, the top left corner of this patch has gray value $a$ , and so on.

1. Looking at this depiction, why are two vanishing points enough to create the perspective foreshortening? Given a camera projection matrix, how would you determine the locations of the vanishing points? What happens if the camera is tilted slightly downwards?

Image /page/7/Figure/5 description: The image shows a black and white drawing of a tall building in two-point perspective. The building is rectangular and has multiple stories, each with several windows. The lines of the building converge towards two vanishing points on the horizon line, which is labeled in the drawing. The vanishing points are located on either side of the building, indicating the perspective technique used. The base of the building is slightly wider than the top, adding to the sense of depth. The drawing is clean and precise, with clear lines and details.

Taken from $w3.joshuanava.biz$

• 1. Given $P(\alpha) = \alpha * T1 + 2 * \alpha * T2 + (1 3 * \alpha) * T3$ , can you provide constraints on to ensure that the point P is inside the triangle (assuming it is non-degenerate, meaning the points form an actual triangle)? Explain your choice. You can, if needed, rely on the test for being inside of a triangle, which we use for a general point on the supporting plane of a triangle.

A. $r = (-0.25, 0.25, 1.0)$ B. $r = (-1.0, 1.0, 1.0)$ C. $r=(-1.0$ D.

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1. Given a robot as illustrated below, your goal is to apply the necessary transformations to move the hands from the original (rest) pose to the target pose.

Image /page/8/Figure/1 description: The image contains two diagrams of stick figures in different poses. The diagram on the left is labeled "rest pose" and shows a stick figure standing straight with arms at its sides. The diagram on the right is labeled "target pose" and shows a stick figure with its arms raised to the sides. The left arm is labeled. The diagrams are used to illustrate the difference between a rest pose and a target pose.

The parts expressed in their original coordinate frame have the following properties:

• $\bullet$ all green parts (arm, forearm and hand) and the body are aligned with the vertical (y) axis
• the shoulders are aligned with the horizontal $(x)$ axis
• all starting points are at the origin, where the starting point of a body part is the first vertex encountered when following the depicted connectivity from the red dot to this body part.

For the target pose, the following properties are given:

• the red point is located at $q$ in $R^2$
• the blue parts are not rotated with regards to the rest pose
• the left arm (depicted on the right see figure) is rotated $45^{\circ}$ clockwise with regards to the left shoulder
• the left forearm is rotated ${60}^{}$ counter-clockwise with regards to the left arm in the target pose

Given the following two transformation matrices:

• $T_p$ : translates by $p$ , where p is a vector in $R^2$
• $R_{\theta}$ : rotates by angle counter-clockwise

Define the sequence of matrices to multiply the points of the hands by in order to draw it at the target position. You should describe two sequences, one for the left hand and one for the right hand. You do not need to provide a single matrix, a product of matrices is fine.

1. We take a photo of the leaning Pisa tower by placing our pinhole camera on the ground at a distance of 10m from the base of the tower (point $p$ ). Consider that the camera centre is at the origin, the camera is oriented along the positive z-axis, the image plane is at $z=2$ , and the ground is the plane $y=0$ .

Point r' is the projection of the top of the leaning tower (point r) and has height equal 1m on the image plane (the exercise uses an unconventionally large camera to ease calculations). Knowing that if the tower stood straight its top would be at point $q$ at a distance of 10m from the ground, what is the distance from the top of the leaning tower to the ground?

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Image /page/9/Figure/0 description: The image shows a diagram with points labeled e, f, p, p', q, r, and r'. There is a coordinate system with axes labeled Y and Z. The point f is labeled with "f = 2". There are lines connecting e to q, e to r, e to p', and e to r'. There are also dashed lines connecting p to r and p' to r'. Additionally, there are vertical lines connecting p to q and p' to r'. The Y-axis is labeled with "1".

$A.5$ B. 6

• $C.7$ $D. 8$
• 1. Given a point light source in $R^3$ at position $l = (0, 4, \sqrt{3})$ and a point on a diffuse surface (no specular component). The normal of the surface at point p is . The light source has the following ambient, diffuse and specular RGB intensities: respectively. Using the Phong Illumination Model, what should the diffuse component of the surface be in order for an observer at position to perceive the colour at point p as ?

A. $k_d = (0.3, 0.4, 0.1)$ B. $k_d = (0.75, 1.0, 0.4)$ C. D.

• 1. Given a point light source at position $L = (-2, 3, 0)$ , and a point $m = (1, 0, 0)$ on a mirror surface with normal $$ . If we rotate the mirror around point m, in what direction should its normal point, such that the light ray is reflected from m towards the point ? Explain your steps to get to the answer (no points is attributed to just writing the final solution vector).

The colour of any point inside one of the squares is defined by bilinear interpolation of the colours of its vertices. Inside the overlapping region, the final colour is defined by adding the interpolated colours of the two squares at point p. Given the following corners: $a = (0, 0), c = (1, 1), e = (0.5, 0.5), g = (1.5, 1.5),$ , compute the resulting colour at p. Explain how you arrived at your solution (just providing the colour of does not count as a valid explanation).

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Image /page/10/Figure/0 description: The image shows two overlapping rectangles, labeled A and B, within a larger gray rectangle. Rectangle A is defined by points a, b, c, and d, while rectangle B is defined by points e, f, g, and h. A point labeled 'p' is located within the overlapping region of the two rectangles.

1. Given a screen of $400 \times 200$ pixels, we want to draw a red rectangle with a blue border. Our rasterization algorithm uses a simple overwrite method when primitives write to the same pixel. Hence, we will first draw a blue rectangle then a smaller red one, giving the result in the image below. The minimum and maximum coordinates of the red rectangle are $(-0.4, -0.75, 0, 1)$ and $(0.6, 0.25, 0, 1)$ , respectively, after the rectangle was projected with the camera matrix. Please indicate the minimum and maximum coordinates of the blue rectangle after projection, each in the form with , if we want a border size of 4 pixels along the edges (see figure below)?

Image /page/10/Figure/2 description: The image shows a rectangular shape with a red fill and a blue border. The border size is indicated by two small black lines with the text 'border size' pointing to each line.

(the figure is for illustration only, it is not supposed to reflect the given coordinates precisely)

1. Given the simplified dart board below formed by two concentric circles with alternating colours. The colour transitions occur precisely at the $x$ and $y$ axis.

Image /page/10/Picture/5 description: The image shows a circular design divided into sections with alternating colors. The design consists of two concentric circles. The outer circle is divided into four quadrants, with the top-left and bottom-right quadrants colored red, and the top-right and bottom-left quadrants colored blue. The inner circle is similarly divided into four quadrants, with the top-left and bottom-right quadrants colored blue, and the top-right and bottom-left quadrants colored red. The overall effect is a symmetrical pattern of alternating red and blue sections within the two circles.

To define the underlying geometry we approximate the outer circle by triangles. We start with four triangles where two edges of each triangle are aligned with the x and y axis, and one vertex is at the centre of the circle. We then subdivide each triangle into two by splitting in half the edge opposite to the centre vertex, obtaining 8 triangles. This subdivision scheme is further repeated many times until the outer circle is well approximated by the triangles, as illustrated below. Given this geometric approximation of the circle by triangles, which of the following textures could be used to achieve our board image above? You should pick the texture with minimum amount of texels that is able to texturize the board, and assume that you can specify new texture coordinates for each triangle at each vertex.

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Image /page/11/Figure/0 description: The image shows a series of circles with lines inside, followed by the letters A, B, C, and D, each followed by a description of a texture and a corresponding image of the texture. The circles have an increasing number of lines inside, starting with four lines and increasing to eight and then twelve. The letters A, B, C, and D are followed by the descriptions "1x2 texture", "1x4 texture", "2x2 texture", and "4x4 texture", respectively. The corresponding images of the textures are made up of red and blue squares. The 1x2 texture is a rectangle with one red square and one blue square. The 1x4 texture is a rectangle with one blue square, one red square, one blue square, and one red square. The 2x2 texture is a square with one blue square, one red square, one red square, and one blue square. The 4x4 texture is a square with one red square, one blue square, one blue square, and one red square.

1. Shadow mapping suffers from a resolution mismatch issue leading to blocky artifacts. We have seen that for color textures mipmapping can avoid some of the texture-related problems. Discuss if this method is useful for shadow mapping. Would you have a suggestion yourself on how to introduce filtering of Shadow-map shadows (short answer)?

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Computer Graphics - $2020/2021$ Resit Examination

1. Given a $512 \times 512$ grayscale image, if you apply a $3 \times 3$ Box filter (averages nine pixels) will the result be the same as applying first a $1 \times 3$ average filter followed by a $3 \times 1$ average filter? Both smaller filters compute the average of three pixels. Explain your answer (only answering yes/no does not lead to any points). You can also ignore any border treatment for your explanation.

You can use the following $3 \times 3$ image patch to help in your explanation. For example, the top left corner of this patch has gray value $a$ , and so on.

Solution: Applying first the horizontal $1 \times 3$ filter the new values of the middle column are: $\begin{array}{l} b' \leftarrow (a+b+c)/3 \\ e' \leftarrow (d+e+f)/3 \end{array}$ Then, applying the vertical filter on the new middle column composed by , and : which is the same as applying the box filter directly over e.

1. Looking at this depiction, why are two vanishing points enough to create the perspective foreshortening? Given a camera projection matrix, how would you determine the locations of the vanishing points? What happens if the camera is tilted slightly downwards?

Image /page/12/Figure/6 description: The image shows a line drawing of a tall building in two-point perspective. The building is rectangular and has multiple stories, each with several windows. Lines converge from the top and bottom of the building towards two vanishing points on the horizon line, illustrating the perspective technique. The horizon line is labeled, and the vanishing points are also labeled.

Taken from $w3.joshuanava.biz$

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Solution: Only two points are needed because the vertical lines are parallel on the image, thus the third vanishing point is projected to infinity.

To determine the locations take the directions of the sides of the building in 3D (with $w = 0$ ) and project using the projection matrix.

If the camera is tilted a third vanishing point would appear at the top (currently, it is at infinity).

1. Given $P(\alpha) = \alpha * T1 + 2 * \alpha * T2 + (1 - 3 * \alpha) * T3$ , can you provide constraints on to ensure that the point P is inside the triangle (assuming it is non-degenerate, meaning the points form an actual triangle)? Explain your choice. You can, if needed, rely on the test for being inside of a triangle, which we use for a general point on the supporting plane of a triangle.

Solution: $\alpha \ast T1 + 2 \ast \alpha \ast T2 + (1 - 3 \ast \alpha) \ast T3 = T3 + \alpha (T1 - T3) + 2 \ast \alpha (T2 - T1)$ Thus and , hence . We can also do it the other way around: Know that a point lies inside of the triangle, when we have We see , thus because of the constraint on a because of the constraint on a because of the constraint on a implies thus . Hence, .

1. When points lie on the same line, they are called collinear. Which of the following points $r$ is collinear to points $p = (2.0, 6.0, 2.0)$ and $q = (3.5, 8.5, 1.0)$ ? All points are expressed in homogeneous coordinates.

A. $r = (-0.25, 0.25, 1.0)$ B. $r = (-1.0, 1.0, 1.0)$ C. $r=(-1.0$ D.

1. Given a robot as illustrated below, your goal is to apply the necessary transformations to move the hands from the original (rest) pose to the target pose.

Image /page/13/Figure/8 description: The image shows two diagrams of a stick figure in different poses. The diagram on the left is labeled "rest pose" and the diagram on the right is labeled "target pose". The stick figure in the "rest pose" diagram has its arms at its sides, while the stick figure in the "target pose" diagram has its arms outstretched. The "target pose" diagram also has a coordinate system labeled X and Y. The diagram on the left has labels for the different body parts, including d_shoulder, d_arm, d_forearm, d_hand, and d_body. The diagram on the right has a label for the left arm.

The parts expressed in their original coordinate frame have the following properties:

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• $\bullet$ all green parts (arm, forearm and hand) and the body are aligned with the vertical (v) axis
• the shoulders are aligned with the horizontal $(x)$ axis
• all starting points are at the origin, where the starting point of a body part is the first vertex encountered when following the depicted connectivity from the red dot to this body part.

For the target pose, the following properties are given:

• the red point is located at $q$ in $R^2$
• the blue parts are not rotated with regards to the rest pose
• the left arm (depicted on the right see figure) is rotated $45^{\circ}$ clockwise with regards to the left $\text{shoulder}$

Given the following two transformation matrices:

• $T_p$ : translates by $p$ , where p is a vector in $R^2$
• $R_{\theta}$ : rotates by angle counter-clockwise

Define the sequence of matrices to multiply the points of the hands by in order to draw it at the target position. You should describe two sequences, one for the left hand and one for the right hand. You do not need to provide a single matrix, a product of matrices is fine.

Solution: left hand: $T(q)T(0,d_{body})T(d_{shoulder},0)R(45)T(0,-d_{arm})R(60)T(0,-d_{forearm})R(-15)$ right hand:

1. We take a photo of the leaning Pisa tower by placing our pinhole camera on the ground at a distance of 10m from the base of the tower (point $p$ ). Consider that the camera centre is at the origin, the camera is oriented along the positive z-axis, the image plane is at $z=2$ , and the ground is the plane $y=0$ .

Point $r'$ is the projection of the top of the leaning tower (point r) and has height equal 1m on the image plane (the exercise uses an unconventionally large camera to ease calculations). Knowing that if the tower stood straight its top would be at point $q$ at a distance of 10m from the ground, what is the distance from the top of the leaning tower to the ground?

Image /page/14/Figure/18 description: The image shows a diagram with points labeled e, f, p, p', q, r, and r'. There is a coordinate system with axes labeled Y and Z. The point f is labeled with the value 2. There are lines connecting the points e to q, e to r, p to q, p' to r', e to r', and p to r. The lines e to r' and p to r are dashed.

$A.5$ $B.6$

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• $C.7$ D. 8
• 1. Given a point light source in $R^3$ at position $l = (0, 4, \sqrt{3})$ and a point on a diffuse surface (no specular component). The normal of the surface at point p is . The light source has the following ambient, diffuse and specular RGB intensities: respectively. Using the Phong Illumination Model, what should the diffuse component of the surface be in order for an observer at position to perceive the colour at point p as ?

A. $k_d = (0.3, 0.4, 0.1)$ B. $k_d = (0.75, 1.0, 0.4)$ C. D.

1. Given a point light source at position $L = (-2, 3, 0)$ , and a point $m = (1, 0, 0)$ on a mirror surface with normal $n=($ . If we rotate the mirror around point m, in what direction should its normal point, such that the light ray is reflected from m towards the point ? Explain your steps to get to the answer (no points is attributed to just writing the final solution vector).

Solution:

normalize light direction $l = (-1/\sqrt{2}, 1/\sqrt{2}, 0)$ normalize desired reflected direction towards Q: add both and normalize again normal direction =

1. Assume a cylinder $x^2 + y^2 = 1$ made out of a mirror material and a light ray r: $r(t) = (2, -1, -2) +$ . In what direction is the reflected light ray leaving? (Hint: Try for the intersection) Calculate the result and explain your solution by commenting on it.

Solution: $ray: (2, -1, -2) + a * (-1, 1, 1)$ For $t=1$ , we have , which is on the cylinder. so the normal there is also The normalized incoming light direction is Hence, the reflected ray is: Thus:

1. Assume two axis-aligned squares in the plane as shown in the image below. Square A has vertices $(a,b,c,d)$ with respective RGB colours $(1,0,0), (0,1,0), (0,0,1), (0,1,0)$ . Square B has vertices with respective colours

The colour of any point inside one of the squares is defined by bilinear interpolation of the colours of its vertices. Inside the overlapping region, the final colour is defined by adding the interpolated colours of the two squares at point p. Given the following corners: $a = (0, 0), c = (1, 1), e = (0.5, 0.5), g = (1.5, 1.5),$ , compute the resulting colour at p. Explain how you arrived at your solution (just providing the colour of does not count as a valid explanation).

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Image /page/16/Figure/0 description: The image shows two overlapping squares, labeled A and B, within a larger rectangular frame. Square A is defined by vertices a, b, c, and d, while square B is defined by vertices e, f, g, and h. A point labeled 'p' is located within the overlapping region of the two squares.

Solution: The vertices have the following weights: $w_a = w_g = 0.25 * 0.25 = 1/16$ The colour of for square A is: The colour of for square B is: The final colour is:

1. Given a screen of $400 \times 200$ pixels, we want to draw a red rectangle with a blue border. Our rasterization algorithm uses a simple overwrite method when primitives write to the same pixel. Hence, we will first draw a blue rectangle then a smaller red one, giving the result in the image below. The minimum and maximum coordinates of the red rectangle are $(-0.4, -0.75, 0, 1)$ and $(0.6, 0.25, 0, 1)$ , respectively, after the rectangle was projected with the camera matrix. Please indicate the minimum and maximum coordinates of the blue rectangle after projection, each in the form with , if we want a border size of 4 pixels along the edges (see figure below)?

Image /page/16/Figure/3 description: The image shows a diagram of a rectangle inside another rectangle. The inner rectangle is red, and the outer rectangle is blue. The space between the two rectangles is labeled as "border size" with arrows pointing to the space on the bottom and right sides. The background is light gray, and there are dotted lines crossing in the center of the image.

(the figure is for illustration only, it is not supposed to reflect the given coordinates precisely)

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Solution: 1 unit horizontally = $2/400 = 1/200 = 0.005$ 1 unit vertically $= 2/200 = 1/100 = 0.01$ $4 \text{ pixels horizontally} = 0.02$ corners of blue rectangle:

1. Given the simplified dart board below formed by two concentric circles with alternating colours. The colour transitions occur precisely at the $x$ and $y$ axis.

Image /page/17/Picture/2 description: The image is a circular design divided into quadrants of alternating blue and red colors. The design features two concentric circles, with the inner circle also divided into quadrants of alternating blue and red. The outer ring, formed between the two circles, follows the same alternating color pattern as the inner circle.

To define the underlying geometry we approximate the outer circle by triangles. We start with four triangles where two edges of each triangle are aligned with the x and y axis, and one vertex is at the centre of the circle. We then subdivide each triangle into two by splitting in half the edge opposite to the centre vertex, obtaining 8 triangles. This subdivision scheme is further repeated many times until the outer circle is well approximated by the triangles, as illustrated below. Given this geometric approximation of the circle by triangles, which of the following textures could be used to achieve our board image above? You should pick the texture with minimum amount of texels that is able to texturize the board, and assume that you can specify new texture coordinates for each triangle at each vertex.

Image /page/17/Figure/4 description: The image shows three circles, each divided into an increasing number of equal parts, followed by an ellipsis. Below the circles are three colored rectangles. The first rectangle, labeled "A. 1x2 texture:", is divided into two equal parts, one red and one blue. The second rectangle, labeled "B. 1x4 texture:", is divided into four equal parts, colored blue, red, blue, and red. The third rectangle, labeled "C. 2x2 texture:", is not shown.

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Image /page/18/Picture/0 description: The image contains two squares, one above the other. Each square is divided into four smaller squares. The top square has the top left and bottom right squares colored red, and the top right and bottom left squares colored blue. The bottom square has the top left and bottom right squares colored blue, and the top right and bottom left squares colored red.

D. $4x4$ texture:

1. Shadow mapping suffers from a resolution mismatch issue leading to blocky artifacts. We have seen that for color textures mipmapping can avoid some of the texture-related problems. Discuss if this method is useful for shadow mapping. Would you have a suggestion yourself on how to introduce filtering of Shadow-map shadows (short answer)?