Properties of Minimal Distance

Abstract

The minimal distance ^m$\bar D$(G, J) between two-dimensional geosets ^gG and ^gJ has important properties, analogous to those of the minimal distance between a pair of one-dimensional sets. Napolitano (1988, 1990) demonstrated that ^m$\bar D$(G, J) is a metric, which implies that

• 1. ^m$\bar D$(G, J) is never negative.

• 2. ^m$\bar D$(G, J) = 0 if, and only if, the two geosets ^gG and ^gJ are equal, so that ^m$\bar D$(G, J) = ^m$\bar D$(G, G) = ^m$\bar D$(J, J); conversely, if ^m$\bar D$(G, J) = 0 then we can conclude that the two geosets are coincident.

• 3. ^m$\bar D$(G, J) = ^m$\bar D$(J, G): when we refer to the simple scheme of shifting elements of one of two geosets in order to make the two geosets coincident, this equality implies that it does not matter whether elements of ^gG are shifted to make ^gG coincident with ^gJ, or elements of ^gJ are shifted to make ^gJ coincident with ^gG.

• 4. If we consider three geosets ^gG, ^gJ, and ^gK and the minimal distance between each pair of them, we find that ^m$\bar D$(G, K) + ^m$\bar D$(K, J) ⩾ ^m$\bar D$(G, J) (14.1 a) ^m$\bar D$(G, J) + ^m$\bar D$(J, K) ⩾ ^m$\bar D$(G, K) (14.1b) ^m$\bar D$(K, G) + ^m$\bar D$(G, J) ⩾ ^m$\bar D$(K, J) (14.1c)