assignment111

assignment111 - p T a 1 . In addition, prove that R T 3 x 3...

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UNIVERSITY OF MINNESOTA DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING 5561 COMPUTER VISION SPRING 2010 ASSIGNMENT 1 (100 Points) Assigned: 2/9/10 Due: 2/18/10 Problem 1 (30 Points) (a) If a xyz = ( - 1 , - 1 , 0) T and b xyz = (1 , 6 , 1) T are the coordinates of two points with respect to the reference coordinate system, determine the corresponding points a uvw and b uvw with respect to the rotated OUVW coordinate system if it has been rotated 30 o about the OX axis and 60 o about the new OZ axis. (b) If a uvw = ( - 1 , - 1 , 0) T and b uvw = (1 , 6 , 1) T are the coordinates of two points with respect to the rotated OUVW coordinate system, determine the corresponding points a xyz and b xyz with respect to the reference coordinate system if it has been rotated 30 o about the OX axis and 60 o about the new OZ axis. Problem 2 (30 Points) Prove that the inverse of matrix T = n x o x a x p x n y o y a y p y n z o z a z p z 0 0 0 1 = " n o a p 0 0 0 1 # = " R 3 x 3 p 0 1 # is given by T - 1 = n x n y n z - p T n o x o y o z - p T o a x a y a z -
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Unformatted text preview: p T a 1 . In addition, prove that R T 3 x 3 = R-1 3 x 3 and R T 3 x 3 R 3 x 3 = I 3 where I 3 is a 3x3 identity matrix. It is known that n 2 x + n 2 y + n 2 z = o 2 x + o 2 y + o 2 z = a 2 x + a 2 y + a 2 z = 1, n T o = n T a = o T a = 0, and n o = a . Problem 3 (10 Points) Determine a matrix T that represents a rotation of angle about the OZ axis, followed by a translation of b units of distance along the old OY axis, followed by a rotation of angle about the new OX axis. Assume frame transforms. Problem 4 (10 Points) Prove that rotation is actually the result of combining scaling and skewing. Use the 3-D transfor-mations. Problem 5 (20 points) What is the shape of the image of a cylinder? Assume perspective projection. 1...
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