Replication of Results - Spatial Mathematics: Theory and Practice through Mapping - page 97

The level of sensitivity to image distortion will vary with the individual;
Figures 4.7a and b may appear the same to some. Figures 4.7a and c will
appear different to more. It is important to have absolute techniques that
guarantee correct answers. Once one knows them, then one can choose when
and when not to violate them.
The reduction of the original by 1/2, producing an image 1/4 of the original
size, may still not be desirable. One can use the unique factorization to build
what is desired. The value of 810 has a number of factors; thus, one might
choose to rescan the image, holding the width at 810 pixels and shaving just
a bit off the height. Now, to create the possibility of various scaling factors,
consider a tiny sliver removed to create a height of 920 = 2*2*2*5*23; there
is an extra factor of 5 that 920 has in common with 810 that 922 did not, so
all of 1/2, 1/5, and 1/10 (1 over 2*5) are scaling factors that will preserve the
aspect ratio. However, one might wish for still more possible scaling factors;
if a slightly larger sliver can be removed, so that the height of the scanned
image is 900 pixels, with 900 = 2*2*3*3*5*5, then 810 and 900 have common
prime power factors of 2, 3, 3*3, and 5, so that the set of scaling factors has
been substantially expanded to include all of:
1/2, 1/3, 1/9, 1/5,
combination of four things one at a time—4!/(1!*3!)
1/6, 1/18, 1/10; 1/27, 1/15; 1/45
combination of four things two at a time—4!/(2!*2!)
1/54, 1/30, 1/90, 1/135
combination of four things three at a time—4!/(3!*1!)
1/270
combination of four things four at a time—4!/(4!*0!)
This set of values for the height, in pixels, should offer a variety of choices
for scaling factors.
4.8 Image security
In creating images, it is useful to have an aspect ratio with the numerator and
the denominator having many common factors; greater flexibility in changing
the image is a consequence, as above. However, there may be situations, par-
ticularly on the Internet where downloading is just a right-click of the mouse
away, in which one wishes to inhibit easy rescaling of an image. When that
is the case, unique factorization via the Fundamental Theorem of Arithmetic
again yields an answer: Choose to scan the image so that the numerator and
the denominator of the aspect ratio are relatively prime—that is, so that the
numerator and the denominator have no factors in common other than 1. In
the case of the 810 by 922 image, altering the aspect ratio from the original
of 810/922 to 810/923 would serve the purpose: 923 is not divisible by any of