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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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