Curvature flows of maximal integral triangulations.(English)Zbl 0947.05017

An integral triangulation (IT) of a planar set $$D$$ is understood to be a triangulation (in the usual sense) such that the vertices of each triangle belong to the integer lattice $$\mathbb{Z}^2$$ and the intersection of any two triangles is an edge or a vertex or empty. (Clearly the boundary of $$D$$ then has to be a polygon whose vertices are lattice points.) Any IT can be refined to a maximal IT (all triangles having the same area $${1\over 2}$$). Take any pair of triangles $$CAB$$, $$C'BA$$ having the common (oriented) edge $$\vec a$$ with root $$A$$ and endpoint $$B$$. If the triangles have the same area then the vector $$(A- C)+(B- C')$$ is obviously parallel to $$A-B$$, hence the latter must be a multiple of the former, with factor $$\varphi(\vec a)$$, say. This gives rise to a real function $$\varphi$$ on the set $$E= E(D)$$ of oriented edges in a maximal IT, called curvature flow on $$E$$, describing the triangulation up to appropriate equivalence. The author proves a sort of discrete Gauss-Bonnet formula: Given any interior vertex $$V$$ with degree $$\deg(V)$$ in a maximal IT, one has the identity $$2(6-\deg(V))= \sum \varphi(\vec a)$$ where the sum is taken over all oriented edges $$\vec a\in E$$ with root $$V$$. Moreover, he gives some local criteria for a function taking integral values on $$E$$ to be the curvature flow of a maximal IT. He also discusses the Farey tree in the present context.

MSC:

 05B45 Combinatorial aspects of tessellation and tiling problems 52C05 Lattices and convex bodies in $$2$$ dimensions (aspects of discrete geometry) 11A55 Continued fractions
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References:

 [1] M. AIGNER, Combinatorial theory, Springer, 1979. · Zbl 0415.05001 [2] H.S.M. COXETER, An introduction to geometry, Wiley, 1989. [3] M. DO CARMO, Differential geometry of curves and surfaces, Prentice Hall, 1976. · Zbl 0326.53001
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