Anyone who has tried to implement algorithms for rotating digitised images stored in
computers will have discovered that it is impossible, except in very simple cases,
such as rotations that are multiples of 90 degrees, to have structure-preserving
combinations of rotations about arbitrarily chosen locations.
E.g. if you have a binary image represented in a 2-D array and you perform rotations
of various amounts about various points in the image you will not in general be able
to reverse those rotations and get back the original image. Even rotation in one
direction by 7 degrees several times around the same point, then performing the same
number of rotations by -7 degrees will not normally restore the image. (Any number
that is not a multiple of 90 will suffice.)
What happens will depend on whether the image is represented in a binary array, or
represented in terms of coordinates and equations for lines, arcs, etc. The rotation
of the binary array will lose structure when a grid location that should be partly
black and partly white after a rotation is forced to be all black or all white, The
rotation of the image stored algebraically will lose structure because coordinates
will be truncated after rotation because of the limited size of the computer. For
example, if a rotation changes the X coordinate of a point to the square root of 2,
then that value cannot be represented by a finite number of bits (using standard
methods for representing real numbers). An example would be rotating a 1 by 1 square
clockwise by 45 degrees about its bottom left corner, based on the coordinate origin.
To illustrate effects of rotations actually performed on a computer, here's an image
drawn using the TGIF drawing package, then saved as a digital image
Here's the result of using TGIF to rotate counter clockwise several times by 7
degrees, then clockwise by the same number of degrees. In this case the intermediate
stages are represented by drawing commands rather than by binary arrays.
Note that corners that previously (approximately) coincided have now moved apart,
(in the figure above) with corresponding changes along the lengths of the lines involved.
Here is the image after clipping the picture to leave the image asymmetrically
located in its frame, then saving as a digitised image.
Below is the result of rotating that clipped image several times about its new centre,
rotating counter-clockwise by 7 degrees then clockwise by the same number of degrees,
using the 'xv' package. I.e. the rotation is done using the digitised image, not
equations and coordinates. In this case the result of the rotations is blurring of
the image. The black surround is a result of copying 'unknown' image contents from
outside the frame during the rotation.
