1 Projective geometry- 2D Acknowledgements Marc Pollefeys: for allowing the use of his excellent slides on this topic http://www. cs . unc . edu /~marc/ mvg / Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision " 04/01/2004 Projective Geometry 2D 2 Homogeneous coordinates 0 = + + c by ax ( ) T a,b,c 0 , 0 ) ( ) ( = + + k kc y kb x ka ( ) ( ) T T a,b,c k a,b,c ~ Homogeneous representation of lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0,0,0) T forms P 2 Homogeneous representation of points 0 = + + c by ax ( ) T a,b,c = l ( ) T y x, x = on if and only if ( )( ) ( ) 0 l 1 1 = = x,y, a,b,c x,y, T ( ) ( ) 0 , 1 , , ~ 1 , , k y x k y x T T The point x lies on the line l if and only if x T l=l T x=0 Homogeneous coordinates Inhomogeneous coordinates ( ) T y x, ( ) T 3 2 1 , , x x x but only 2DOF 04/01/2004 Projective Geometry 2D 3 Points from lines and vice-versa l' l x = Intersections of lines The intersection of two lines and is l l' Line joining two points The line through two points and is x' x l = x x' Example 1 = x 1 = y 04/01/2004 Projective Geometry 2D 4 Ideal points and the line at infinity ( ) T 0 , , l' l a b = Intersections of parallel lines ( ) ( ) T T and ' , , l' , , l c b a c b a = = Example 1 = x 2 = x Ideal points ( ) T 0 , , 2 1 x x Line at infinity ( ) T 1 , 0 , 0 l = = l 2 2 R P Note that in P 2 there is no distinction between ideal points and others Note that this set lies on a single line, 04/01/2004 Projective Geometry 2D 5 Summary The set of ideal points lies on the line at infinity, intersects the line at infinity in the ideal point A line parallel to l also intersects in the same ideal point, irrespective of the value of c’. In inhomogeneous notation, is a vector tangent to the line. It is orthogonal to (a, b) -- the line normal. Thus it represents the line direction. As the line’s direction varies, the ideal point varies over . --> line at infinity can be thought of as the set of directions of lines in the plane. 04/01/2004 Projective Geometry 2D 6 A model for the projective plane exactly one line through two points exaclty one point at intersection of two lines Points represented by rays through origin Lines represented by planes through origin x1x2 plane represents line at infinity
4
Embed
Homogeneous coordinates Projective geometry- 2D ax+by+ …3 04/01/2004 Projective Geometry 2D 13 Projective transformations A projectivity is an invertible mapping h from P2 to itself
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Projective geometry- 2D
Acknowledgements
Marc Pollefeys: for allowing the use of his excellent slides on this topichttp://www.cs.unc.edu/~marc/mvg/
Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision"
04/01/2004 Projective Geometry 2D 2
Homogeneous coordinates
0=++ cbyax ( )Ta,b,c
0,0)()( =++ kkcykbxka ( ) ( )TTa,b,cka,b,c ~
Homogeneous representation of lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3 (0,0,0)T forms P2