Wednesday, February 9, 2011

Quick questions

Given an arbitrary 3x3 matrix, how do you know it's a homography?

Given an arbitrary 3x3 matrix, how do you know it's an affine transformation?

Given an arbitrary 3x3 matrix, how do you know it's an Euclidean transformation?



Answers:
1) Should be invertible.
2) Should be invertible and last row should be (0,0,1).
3) Should be invertible and last row should be (0,0,1) and upper-left 2x2 matrix should be orthogonal.


How do you know it's invertible?

How do you know the upper-left 2x2 matrix is orthogonal?


Answers
1) determinant should be zero.
2) That means its columns ( & hence rows too) should be orthonormal. Check their dot product.