αβ-Triangulation on the Euclidean Plane
Vyacheslav Rustamyan · Geometry & Graphics · 2025
In the framework of solving the problem of approximating free-form surfaces by polyhedra with groups of congruent faces, an idea was proposed to predefine one or more groups of congruent triangles for any triangulation without limiting the flexibility of subsequent optimization. This led to the creation of the concept of αβ-triangulation. The article reveals theoretical aspects of αβ-triangulation in two-dimensional Euclidean space E2 , based on the intersection of elements from set theory, graph theory, and combinatorial topology. The paper presents a detailed study of this new mathematical model. The author introduces a precise definition of the notion of αβ-triangulation, formulates its main properties, and establishes important operations such as cutting and sewing that allow efficient transformation of the triangulation structure. A detailed description is given of the algorithm for constructing an αβ-triangulation from an arbitrary strongly connected triangulation, providing a universal approach to forming optimal structures for specific tasks. Furthermore, proof of consistency and independence of the introduced system of axioms is provided, which significantly strengthens the theoretical foundations of the model. Theoretical development of αβ-triangulation opens up broad prospects for solving problems in computational geometry, offering an effective tool for representing and processing complex forms of spatial objects. However, before practical application of this model, additional experiments and analysis of its efficiency compared to existing analogs are necessary. Thus, further research will determine the role and significance of the proposed model within modern technologies and methods of geometric modeling.