# A Distance Function for Comparing Straight-Edge Geometric Figures

• Apoorva Honnegowda Roopa
• Shrisha Rao
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 730)

## Abstract

This chapter 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.

## Keywords

Geometric similarity Iterative Proportional Fitting Procedure Euclidean distance

## 2010 Mathematics Subject Classification.

65D10 (primary) 51K05 (secondary)

## References

1. 1.
Stephen Stoker, H.: General, organic, and biological chemistry. Cengage Learn. (2009)Google Scholar
2. 2.
Bourne, P.E., Gu, J.: Structural bioinformatics, 2 ed., Wiley (2009)Google Scholar
3. 3.
Gillespie, R.J., Robinson, E.A.: Models of molecular geometry. Chem. Soc. Rev. 34, 396–407 (2005)Google Scholar
4. 4.
Sen, A., Chebolu, V., Rheingold, A.L.: First structurally characterized geometric isomers of an eight-coordinate complex. structural comparison between cis- and trans-diiodobis(2,5,8-trioxanonane)samarium. Inorg. Chem. 26(11), 1821–1823 (1987)
5. 5.
Stashans, A., Chamba, G., Pinto, H.: Electronic structure, chemical bonding, and geometry of pure and Sr-doped CaCO3. J. Comput. Chem. 29(3), 343–349 (2008)
6. 6.
Zaka, B.: Theory and applications of similarity detection techniques, Ph.D. thesis, Institute for Information Systems and Computer Media (IICM), Graz University of Technology, Graz, Austria, 2 2009Google Scholar
7. 7.
Donald, S.B.: The perception of similarity. In: Robert, G.C. (ed.) Avian Visual Cognition, September 2001Google Scholar
8. 8.
Michael, J.T., Bülthoff, H.H.: Image-based object recognition in man, monkey and machine. Cognition 67(1–2), 1–20 (1998). doi: Google Scholar
9. 9.
Vermorken, M., Szafarz, A., Pirotte, H.: Sector classification through non-gaussian similarity. Appl. Financ. Econ. 20(11) (2008). doi:
10. 10.
Sweeney, C., Kneip, L., Höllerer, T., Turk, M.: Computing similarity transformations from only image correspondences. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2015), pp. 3305–3313, June 2015. doi:
11. 11.
Shiuh-Sheng, Y., Liou, J.-R., Shen, W.-C.: Computational similarity based on chromatic barycenter algorithm. IEEE Trans. Consum. Electron. 42(2), 216–220 (1996)
12. 12.
Komosinski, M., Koczyk, G., Kubiak, M.: On estimating similarity in artificial and real organisms. Theor. Biosci. 120(3–4), 271–286 (2001)
13. 13.
Komosinski, M., Kubiak, M.: Quantitative measure of structural and geometric similarity of 3D morphologies. Complexity 16(6), 40–52 (2011)
14. 14.
Heller, V.: Scale effects in physical hydraulic engineering models. J. Hydraul. Res. 49(3), 293–306 (2011)Google Scholar
15. 15.
Pallett, G.: Geometric similarity–some applications in fluid mechanics. Educ. Train. 3(2), 36–37 (1961)
16. 16.
Ullmann, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)
17. 17.
Grewal, B.S., Grewal, J.S.: Higher Engineering Mathematics, 40th edn. Khanna Publishers, New Delhi (2007)Google Scholar
18. 18.
Wong, D.W.S.: The reliability of using the iterative proportional fitting procedure, 340–348 (1992)Google Scholar
19. 19.
Lahr, M., de Mesnard, L.: Biproportional techniques in input-output analysis: table updating and structural analysis. Econ. Syst. Res. 16(2), 115–134 (2004)
20. 20.
Edwards Deming, W., Stephan, F.F.: On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11(4), 427–444 (1940)Google Scholar
21. 21.
Rudin, W.: Principles of Mathematical Analysis. McGrawHill Inc., New York (1976)