Universal tools for analysing structures and interactions in geometry


  • Besim Berisha Haxhi Zeka University
  • Xhevdet Thaqi Public University “Kadri Zeka”
  • Astrit Ramizi Public University “Kadri Zeka”
  • Agon Mahmuti Public University “Kadri Zeka”



Palabras clave:

geometric transformations, symmetry, perspective, invariance, Möbius transform


This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions.

Biografía del autor/a

Besim Berisha, Haxhi Zeka University

Faculty of Businesses

Xhevdet Thaqi, Public University “Kadri Zeka”

Faculty of Applied Sciences

Astrit Ramizi, Public University “Kadri Zeka”

Faculty of Applied Sciences

Agon Mahmuti, Public University “Kadri Zeka”

Faculty of Applied Sciences


Barroso-Laguna A, Brachmann E, Prisacariu VA, Brostow GJ, Turmukhambetov D. Two-view geometry scoring without correspondences. Comp Vision Pattern Recogn. 2023. https://doi.org/10.48550/arXiv.2306.01596.

Buzzelli M. Angle-retaining chromaticity and color space: Invariants and properties. J Imag. 2022;8(9):232. https://doi.org/10.3390/jimaging8090232.

Leopold C. Symmetry concepts as design approach. Aesthetics between irregularity and regularity. In: 12th SIS-Symmetry Congress “Symmetry: Art and Science” (202-209). Budapest: International Society for the Interdisciplinary Study of Symmetry, 2022. https://doi.org/10.24840/1447-607X/2022/12-25-202.

Bense, M. Aesthetica: Einführung in die neue Aesthetik. Agis: Verlag, 1982. Available at: https://monoskop.org/images/7/7c/Bense_Max_Aesthetica_Einfuehrung_in_die_neue_Aesthetik_1965.pdf.

Gözütok U, Anıl Çoban H, Sagiroglua Y. A differential approach to detecting projective equivalences and symmetries of rational 3D curves. Inf Organiz. 2023;33(1):100450. https://doi.org/10.1016/j.infoandorg.2023.100450.

Sergeant-Perthuis G, Rudrauf, D, Ognibene D, Tisserand Y. Action of the Euclidean versus Projective group on an agent’s internal space in curiosity driven exploration: A formal analysis. Artific Intellig. 2023. https://doi.org/10.48550/arXiv.2304.00188.

Wolfson HJ. Model-based object recognition by geometric hashing. In: Proceedings of the 1st European Conference on Computer Vision (526-536). Berlin: Springer, 1990. https://doi.org/10.1007/BFb0014902.

Gomes H. Poincaré invariance and asymptotic flatness in Shape Dynamics. Gener Relat Quant Cosmol. 2012. https://doi.org/10.48550/arXiv.1212.1755.

Weyl H. Symmetry. Princeton: Princeton University Press, 2017. Available at: https://www.scirp.org/(S(lz5mqp453edsnp55rrgjct55.))/reference/referencespapers.aspx?referenceid=1120383.

Royer FL. Detection of symmetry. Journal of Experimental Psychology: Human Percep Perform. 1981;7(6):1186-1210. https://doi.org/10.1037//0096-1523.7.6.1186.

Gros P, Quan L. Projective invariants for vision. 1992. Available at: https://inria.hal.science/inria-00590013/PDF/invariant.pdf.

Thaqi X. Learn to teach geometric transformations. Barcelona: University of Barcelona, 2009.

Thaqi X, Aljimi E. The structure of n harmonic points and generalization of Desargues’ theorems. Mathematics. 2021;9(9):1018. https://doi.org/10.3390/math9091018.

Patterson BC. Projective geometry. New York: John Wiley & Sons, 1937.

Liberti L, Lavor C, Maculan N, Mucherino A. Euclidean distance geometry and applications. SIAM Rev. 2014;56(1):3-69. https://doi.org/10.1137/120875909.

Olsen J. The geometry of Möbius transformations. Rochester: University of Rochester, 2010. Available at: https://johno.dk/mathematics/moebius.pdf.

Flusser J, Lébl M, Šroubek F, Pedone M, Kostková J. Blur invariants for image recognition. Int J Comput Vis. 2023;131:2298-2315. https://doi.org/10.1007/s11263-023-01798-7.

Khadjiev D, Ören İ. Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space. Glob Invar Paths Curve. 2019;27(2):37-65. https://doi.org/10.2478/auom-2019-0018.

Kanbak C, Moosavi-Dezfooli SM, Frossard P. Geometric robustness of deep networks: analysis and improvement. 2018. Available at: https://openaccess.thecvf.com/content_cvpr_2018/papers/Kanbak_Geometric_Robustness_of_CVPR_2018_paper.pdf.

Berton G, Masone C, Paolicelli V, Caputo B. Viewpoint invariant dense matching for visual geolocalization. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) (12169-12178). Montreal: IEEE, 2021. https://doi.org/10.48550/arXiv.2109.09827.

Bayro-Corrochano E. A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995-2020. Digit Obj Ident. 2021;9:104326-104355. https://doi.org/10.1109/ACCESS.2021.3097756.

Velich R, Kimmel R. Deep signatures – Learning invariants of planar curves. Comp Vis Pattern Recognit, 2022. https://doi.org/10.48550/arXiv.2202.05922.

Zhao C, Yang J, Xiong X, Zhu A, Cao Z, Li X. Rotation invariant point cloud analysis: Where local geometry meets global topology. Pattern Recognition. 2022;127:108626. https://doi.org/10.48550/arXiv.1911.00195.

Rudrauf D, Bennequin D, Williford K. The Moon illusion explained by the projective consciousness model. J Theor Biol. 2020;507:110455. https://doi.org/10.1016/j.jtbi.2020.110455.

Gorda E, Serdiuk A, Nazarenko, I. Determining the invariant of inter-frame processing for constructing the image similarity metric. East-European J Ent Tech. 2023;2(122):19-25. https://doi.org/10.15587/1729-4061.2023.276650.




Cómo citar

Berisha, B., Thaqi, X., Ramizi, A., & Mahmuti, A. (2023). Universal tools for analysing structures and interactions in geometry. Innovaciencia, 11(1). https://doi.org/10.15649/2346075X.3602



Artículo de investigación científica y tecnológica



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