Moskva, Moscow, Russian Federation
Russian Federation
This paper continues the study of αβ-triangulation a triangulation with a designated set of α-triangles that is optimal in the sense of minimising the total length of β-edges. The model was previously defined on the Euclidean plane and extended to closed spherical surfaces; a generalisation to arbitrary genus g has not been obtained until now. The research tool is the Euler – Poincaré formula combined with the characteristic identity V = 3a for αβ-triangulation. We obtain formulas for the number of β-faces b = 5a + 4g – 4 and β-edges E β = 6(a + g – 1) of a closed αβ-triangulation of genus g. The results are extended to compact orientable surfaces with an arbitrary number of boundary components h: an exact equality for b, a double inequality for b in terms of (a, g, h), and the necessary existence condition a ≥ h. All previously known results — for the sphere, for the closed surface of genus and for the topological disk — follow as special cases. The formulas are verified on 15 closed αβ-triangulations of genera 1, 2, 4 (up to a = 2884) and on eight samples of disks with holes. The results allow predicting the size of an αβ-triangulation before its construction, verifying the correctness of generating algorithms, and choosing discretisation parameters when approximating free-form surfaces by polyhedra with groups of congruent faces.
αβ-triangulation, Euler characteristic, closed surfaces, surfaces with boundary, discrete surfaces
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