hw2_line2line

# hw2_line2line - ECE 181b Homework 2 Spring 2006 1 Image of...

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ECE 181b Homework 2 Spring 2006 1 Image of Line in 3D Question 1 Show Analytically that the image of a line in 3-D space is a line in the image. Assume perspective projection. 10 points Answer 1 The equation of a straight line in 3D is x = m x z + b x (1) y = m y z + b y . Without loss of generality (wlog), we assume that the center of projection is at the origin. Upon projection onto the plane z = f , the image of the point ( x 0 , y 0 , z 0 ) is x i = ( f/z 0 ) x 0 (2) y i = ( f/z 0 ) y 0 . Combining the two sets of equations (1) and (2), we get x i z 0 /f - m x z 0 = b x (3) y i z 0 /f - m y z 0 = b y . (4) Dividing equation 3 by equation 4, we get x i /f - m x y i /f - m y = b x b y x i = b x b y y i + f ˆ m x - b x m y b y ! x i = my i + b. Hence, the image of a straight line in 3D is also a straight line. 1

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2 Vanishing Point Question 2 Straight lines in the three-dimensional world are projected as straight lines into the two-dimensional image (see problem 1 above). How- ever, the projections of parallel lines intersect at a vanishing point. Where
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