CS480/CS680 Problem Set 1 Due in class Tuesday, February 12 at the beginning of lecture. Please prepare the answers to these questions, neatly written or typed, on separate paper. 1. (a) (3 points) Write a 4 × 4 homogeneous transform matrix M that when applied to a point ( x,y,z, 1) yields ( x0 ,y0 ,z0 ,w0 ) where x0 =-1 √ 2 x-1 √ 2 z + a y0 =-2 y z0 = 1 √ 2 z-1 √ 2 x + a w0 = 1 (b) (12 points)In words , what four basic computer graphics transforms occur when we apply M to a 3D point? Give a homogeneous transform matrix for each, and show the order in which they are multiplied. 2. (15 points) We have a unit cube centered at the point c = (1 . 5 ,0 . 5 , 2 . 5) . Derive the ho-mogeneous transformation matrix that will rotate the cube by angle θ around a vector in the direction v = (-1,0,-1). The pivot point for the rotation is the cube’s center c . 3. Use quaternions in your answers to the following. • (5 points) Prove that in general two 3D rotations about different rotation axes do not
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