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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern