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Coordinate transformations

The coordinate transformation of interest here is the so-called ``homographic'' coordinate transformation, given by y=(Ax+b)/(cx+1), where x is the coordinate of the original image (x is a vector containing components x1 and x2), the range, and where y is the coordinate of the destination image, the domain. A is a 2 by 2 matrix, b is a 2 by 1 vector, and c is a 1 by 2 vector.

Examples of homographic coordinate transformations can be found on a page called Periodicity from a new perspective

The coordinate transformation y=Ax+b is the more familiar affine mapping that maps rectangles to parallelograms and preserves periodicity. The affine coordinate transformation is a special case of the homographic coordinate transformation

In addition to not preserving parallel lines, the homographic coordinate transformation also does not preserve periodicity: uniformly spaced objects may become "chirped", and so the c-vector in the denominator of the coordinate transformation is known as the "chirp-rate".

Some people refer to coordinate transformations as "warps", but this term is misleading for the particular homographic coordinate transformation in particular, because one of the most important properties of the homographic coordinate transformation is that it preserves straight lines. Many coordinate transformations cause straight lines to bend ("warp") but the homographic coordinate transformation does not.