 # Graphs Worksheet

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ml951837099

Mathematics

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Homework 2 - Transformations 1. Using the labels Translate, Rotation, Scale, or Reflect, label the transformation of point p when multiplied by matrix M (p’ = Mp): (a)   0 −1 M= 1 0 (b) " √ M= 2 2√ √ # 2 √2 2 2 0 4 0 0 0 0 2 0 − 22 (c)  2 0 M = 0 0  0 0  0 1 (d)  −1 0 M= 0 1  (e)   1 0 3 M = 0 1 2 0 0 1 2. Find the matrix for a 120◦ rotation about the axis defined by the vector r = (1,1,0). 3. Match the following 2D homogeneous matrices to the transformations in the image:  √  √     −√ 23 − √21 0 1.5 0 0 1 0 4   a.  1 − 3 0 b.  0 −1 0 c. 0 1 1 2 2 0 0 1 0 0 1 0 0 1 1 4. Describe a sequence of Translate(x,y), Rotate(degrees), and/or Scale (by a value of ) that when multiplied describe the below transformation to go from the ’Before’ to ’After’. 5. (a) Describe (using a sequence of (3 x 3) matrix multiplications) the transformations needed to transform the triangle from A to B in the figure below: (a)         0  x x 4 ∗ ∗ ∗ y   y0  =  B w w0 2 A 0 0 2 4 6 8 (b) Using the transformations found in part (a), multiply the following points with the matrix. P1 .(4, 1)P2 .(4, 3) 2 Homework 2 - Modeling 1. Define uniform, attribute and varying variables. 2. What type is the shader and describe its function? (a) attribute vec2 a_Position; main() { gl_Position = vec4(0.5 * a_Position, 0.0, 1.0); } (b) main() { gl_FragColor = vec4(0.0, 0.0, 1.0, 1.0); } 3. Using linear interpolation, calculate the color of the following points that are on the line defined by the 2D points p1 = [3,5] with color [255,0,0] and p2 = [10, 5] with color [0,0,255]. (a) [4.8, 5] (b)[7.5, 5] (b)[9, 5] 4. Using barycentric coordinates, calculate the color for each of the following points inside the triangle defined by points p1 = [0,0,0], p2 = [3,5,0] and p3 = [6,0,0] with respective RGB colors c1 = [255, 0, 0], c2 = [0, 255, 0], c3 = [0, 0, 255]. (a) [1, 1, 0] (b) [3, 4, 0] (c) [5, 0.25, 0] 5. Calculate the RGB color value for point P in the diagram below using bilinear interpolation. Show the intermediate values that you compute for interpolating along one of the two axes. The colors of the points A, B, C and D are the following: A = RGB(0, 0, 0) B = RGB(200, 200, 200) C = RGB(200, 0, 0) D = RGB(200, 0, 0) 1 6. Given a polygon with vertices P1, P2, P3, each represented by a 3D vector [x y z], what formula calculates the face’s normal? 2
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