A Distance Function for Comparing Straight-Edge Geometric Figures

TL;DR

This paper introduces a novel distance function for comparing planar straight-edge geometric figures, combining angular dissimilarity and edge length disproportionality, which satisfies all distance properties and is computationally efficient.

Contribution

The paper presents a new distance function for geometric figures that integrates angular and edge length differences, ensuring metric properties and computational simplicity.

Findings

Function satisfies all properties of a distance metric.

Distance computed via projections and IPFP for best fit line.

Demonstrated behavior on sample geometric figure pairs.

Abstract

This paper defines a distance function that measures the dissimilarity between planar geometric figures formed with straight lines. This function can in turn be used in partial matching of different geometric figures. For a given pair of geometric figures that are graphically isomorphic, one function measures the angular dissimilarity and another function measures the edge length disproportionality. The distance function is then defined as the convex sum of these two functions. The novelty of the presented function is that it satisfies all properties of a distance function and the computation of the same is done by projecting appropriate features to a cartesian plane. To compute the deviation from the angular similarity property, the Euclidean distance between the given angular pairs and the corresponding points on the $y=x$ line is measured. Further while computing the deviation from the edge length proportionality property, the best fit line, for the set of edge lengths, which passes through the origin is found, and the Euclidean distance between the given edge length pairs and the corresponding point on a $y=mx$ line is calculated. Iterative Proportional Fitting Procedure (IPFP) is used to find this best fit line. We demonstrate the behavior of the defined function for some sample pairs of figures.