Mapping Multidimensional Space to One Dimension for Computer Output Display

Abstract—The problem considered is the mapping of a functional's multidimensional bounded domain onto a bounded interval of the real line. This transformation is to have the property that "neighboring" points in the bounded interval are necessarily mapped from "neighboring" points in the bounded domain. The multidimensional bounded domain is partitioned into a finite number of elementary regions while the bounded interval is partitioned into the same finite number of elementary intervals. A one-to-one correspondence is defined between the elementary regions and intervals such that neighboring elementary intervals have corresponding multidimensional elementary regions that are neighboring. The degree of neighborliness is controlled by controlling the fineness of partitioning. By suitably defining an equivalent function over the bounded interval, properties of the original functional can be displayed.