Criteria for Visualizable Mathematical Structures
Keywords:
Visualizable mathematical structures, Computer graphics modeling, Geometric representation, Vector geometry, Computational visualization, Mathematical modeling
Abstract
The relationship between abstract mathematical structures and their graphical representation constitutes a fundamental problem in computer graphics. This paper examines the criteria that allow a mathematical structure to be transformed into a visual form while preserving its intrinsic properties. Emphasis is placed on geometric interpretation, computational realizability, and structural stability. Drawing on principles of vector geometry and spatial modeling, a coherent framework is proposed for identifying visualizable structures. The paper further incorporates mathematical examples and schematic descriptions to clarify the transition from formal definition to graphical representation. The results contribute to both theoretical understanding and practical modeling methodologies.
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References
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3. Cavallo, M. (2021). Higher dimensional graphics: Conceiving worlds in four spatial dimensions and beyond. In Computer Graphics Forum (Vol. 40, No. 2, pp. 51-63).
4. Cohen, T. S., & Welling, M. (2014). Transformation properties of learned visual representations. arXiv preprint arXiv:1412.7659.
5. Foley, J. D., van Dam, A., Feiner, S. K., & Hughes, J. F. (1996). Computer graphics: Principles and practice (2nd ed.). Addison-Wesley.
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8. Hearn, D., & Baker, M. P. (2014). Computer graphics with OpenGL (4th ed.). Pearson.
9. Hoyrup, M. (2014). Irreversible computable functions. In STACS-31st Symposium on Theoretical Aspects of Computer Science-2014.
10. Hughes, J. F., van Dam, A., McGuire, M., Sklar, D. F., Foley, J. D., Feiner, S. K., & Akeley, K. (2014). Computer graphics: Principles and practice (3rd ed.). Addison-Wesley.
11. Janke, S. J. (2014). Mathematical structures for computer graphics. Wiley.
12. Jia, W., Sun, M., Lian, J., & Hou, S. (2022). Feature dimensionality reduction: a review. Complex & Intelligent Systems, 8(3), 2663-2693.
13. Kraak, M. J. (2013). Cartography: visualization of spatial data. Routledge.
14. Kreyszig, E. (2011). Advanced engineering mathematics (10th ed.). Wiley.
15. Lorensen, W. E., & Cline, H. E. (1987). Marching cubes: A high resolution 3D surface construction algorithm. ACM SIGGRAPH Computer Graphics, 21(4), 163–169. https://doi.org/10.1145/37402.37422
16. Mancosu, P. (2005). Visualization in logic and mathematics. In Visualization, explanation and reasoning styles in mathematics (pp. 13-30). Dordrecht: Springer Netherlands.
17. Meinardus, G. (2012). Approximation of functions: Theory and numerical methods. Springer Science & Business Media.
18. Mortenson, M. E. (2006). Geometric modeling (3rd ed.). Industrial Press.
19. Munzner, T. (2014). Visualization Analysis and Design.
20. Pharr, M., Jakob, W., & Humphreys, G. (2016). Physically based rendering: From theory to implementation (3rd ed.). Morgan Kaufmann.
21. Ren, Y., & Wei, G. W. (2025). Interpretability and Representability of Commutative Algebra, Algebraic Topology, and Topological Spectral Theory for Real‐World Data. Advanced Intelligent Discovery, e202500207.
22. Robinson, R. C. (2012). An introduction to dynamical systems: continuous and discrete (Vol. 19). American Mathematical Soc..
23. Rogers, D. F. (2001). An introduction to NURBS: With historical perspective. Morgan Kaufmann.
24. Salomon, D. (2012). Computer graphics and geometric modeling. Springer Science & Business Media.
25. Shirley, P., Marschner, S., & others. (2009). Fundamentals of computer graphics (3rd ed.). A K Peters.
26. Ware, C. (2021). Information Visualization: Perception for Design.
Published
2026-06-15
How to Cite
Papadopoulos, E., & Kougias, I. (2026). Criteria for Visualizable Mathematical Structures. European Scientific Journal, ESJ, 54, 446. Retrieved from https://eujournal.org/index.php/esj/article/view/21167
Section
ESI Preprints
Copyright (c) 2026 Evanthios Papadopoulos, Ioannis Kougias
This work is licensed under a Creative Commons Attribution 4.0 International License.
