Use of several non-Euclidean metrics to compute distances between every two points in a plane bounded convex set

Abstract

We consider movements between two chosen points in a plane-bounded set. The length of the movement between two points is measured by the Euclidean metric. We can then obtain the distribution of distances regarding the bounded set by repeatedly measuring the lengths of the movements. When we select two points uniformly in a bounded set, the shape of the bounded set affects the distribution. The distribution is easily computable by integral approximation and usefully remains invariant under the group of congruence transformations on the bounded set. For these reasons, some fundamental methods based on this distribution have been used in shape analysis. Thus, this distribution is important to some practical methods based on the Euclidean metric. Although some articles have been devoted to the study of non-Euclidean metrics in pattern recognition including shape analysis, only a few attempts have so far been made using the distribution of such non-Euclidean metrics. In this paper, we present the expression for the distributions with several non-Euclidean metrics using a density for the set of lines. This expression can be efficiently calculated as a sum of distributions, by generating a finite set of lines. We concentrate on a bounded convex set in the plane to define a non-Euclidean metric. In examples of such metrics on the convex set, we discuss those well-known in taxicab and projective geometries. In the experimental results using several convex sets, we visualize the distributions with the non-Euclidean metrics by plotting their graphs. Comparing the distribution based on the Euclidean metric with those based on the non-Euclidean metrics, we reveal several differences among the distributions.