#### 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.