Lecture 4: Projective Geometry

plane projective transformation
(Z_w = 0 -> 3-by-3 matrix representing a general plane to plane projective transformation)
http://en.wikipedia.org/wiki/Projective_plane

rotation about the optical center
: depth-independent transformation
(H does not depend on 3D structure.)

synthetic rotations
projective warping
http://en.wikipedia.org/wiki/Image_warping

null space
http://en.wikipedia.org/wiki/Kernel_(matrix)
The null space of a matrix with n columns is a linear subspace of n-dimensional Euclidean space.
The null space of A is the same as the solution set to the homogeneous system.
(cf. http://en.wikipedia.org/wiki/Kernel_(mathematics)
http://en.wikipedia.org/wiki/Kernel_(linear_operator) )


> 2d transformation
- Euclidian transformation
- affine transformation
http://en.wikipedia.org/wiki/Affine_transformation
- similarity transformation
http://en.wikipedia.org/wiki/Similarity_transformation
- projective transformation
http://en.wikipedia.org/wiki/Projective_transformation


> projective plane
homography
http://en.wikipedia.org/wiki/Homography

homogeneous coordinates
http://en.wikipedia.org/wiki/Homogeneous_coordinate

perspective projection
http://en.wikipedia.org/wiki/Perspective_projection#Perspective_projection

projective space
http://en.wikipedia.org/wiki/Projective_space

A point in the image is a ray in projective space.

=> All points on the ray are equivalent.

3D Projective Geometry

These concepts generalize naturally to 3D
- Homogeneous coordinates
Projective 3D points have four coords:  X = (X,Y,Z,W)
- Duality
A plane L is also represented by a 4-vector
Points and planes are dual in 3D: LTP = 0
- Projective transformations
Represented by 4x4 matrices T:  X’ = TX,    X’ = L X -1

However
- Can’t use cross-products in 4D.  We need new tools
Grassman-Cayley Algebra
: generalization of cross product, allows interactions between points, lines, and planes via “meet” and “join” operators
- Or just use inhomogeneous representation in 3D.

http://en.wikipedia.org/wiki/Grassman_algebra


vanishing point
http://mathdl.maa.org/mathDL/1//?pa=content&sa=viewDocument&nodeId=477&bodyId=598

vanishing line

http://en.wikipedia.org/wiki/Perspective_(graphical)