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Elementary notions of lattice trigonometry

Oleg Karpenkov
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Abstract

In this paper we study properties of lattice trigonometric functions of lattice angles in lattice geometry. We introduce the definition of sums of lattice angles and establish a necessary and sufficient condition for three angles to be the angles of some lattice triangle in terms of lattice tangents. This condition is a version of the Euclidean condition: three angles are the angles of some triangle iff their sum equals π. Further we find the necessary and sufficient condition for an ordered n-tuple of angles to be the angles of some convex lattice polygon. In conclusion we show applications to theory of complex projective toric varieties, and a list of unsolved problems and questions.

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Oleg Karpenkov. 2007-07-30. Elementary notions of lattice trigonometry. https://arxiv.org/abs/math/0604129

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