2.3 Mapping cyberspace over a visualization media
Suppose that the goal of the transformation has a central feature of obtaining a monotone relationship between distances. Then only the rank order of the dissimilarities has to be preserved by the transformation. Hence, the metric is abandoned during the mapping. Therefore the transformation must obey the monotonicity constraint
monotonicity not satisfied
If the metric nature of the transformation is to be preserved, a configuration will have to satisfy
where is a monotonic function of the distance; a possible straighforward example could be
The stronger constraint that we can put on the mapping is the isometry, having then a perfect preservation of the topology. An isometry is defined by
It is obvious that a mapping that satisfies the isometry also conserves the monotonicity among distances.