The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius
Résumé
In this paper, we introduce the Tri-Quarter Topological Duality Theorem, the foundation of a novel mathematical framework that unifies complex, Cartesian, and polar coordinate systems on the complex plane $\mathbb{C}$ while equipping the circle $T_r$ of radius $r > 0$ with a new topological property. Our framework integrates a generalized coordinate system---where real and imaginary components are assigned unique phase pairs---with a structured orientation that elevates $T_r$ to an active separator with intrinsic directional properties. We prove that $T_r$, as the boundary zone, exhibits topological duality with the inner zone $X_{-,r}$ ($||\vec{x}|| < r$) and outer zone $X_{+,r}$ ($||\vec{x}|| > r$), ensuring consistent separation between inner and outer radial directions across $T_r$ with a phase pair map encoding additional information. We also introduce the Escher Tri-Quarter Reflective Duality Theorem, proving reflective duality across $T_r$ via a circle inversion map that preserves phase pairs while swapping $X_{-,r}$ and $X_{+,r}$. Moreover, these phase pair assignments provide a combinatorial classification of directional structures in $\mathbb{C}$, enhancing the topological analysis. This approach offers insights into topological separation, orientation, and reflection, facilitating analysis of systems with circular symmetry, with potential applications in fields such as black hole physics, signal processing, and other areas reliant on complex domain partitioning. A case study on quadrant-based transformations demonstrates streamlined directional mappings, geometric elegance, unified classification, and computational efficiency in $\mathbb{C}$. For instance, in the case study, the Tri-Quarter approach consistently reduces the number of comparison conditional checks from a maximum of 7 in the standard method to a maximum of 4, significantly enhancing computational efficiency. A software tool visualizes some of these concepts, with future work aimed at exploring practical implementations.
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |
![]()
Est identique à 10.36227/techrxiv.174352256.60328355/v3 Autre Schmidt, N. O. (2025). The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius. Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.36227/techrxiv.174352256.60328355/v3
Also available on TechRxiv with DOI: 10.36227/techrxiv.174352256.60328355/v3