Elementary notions of lattice trigonometry

TL;DR

This paper explores properties of lattice trigonometric functions and angles, establishing conditions for lattice triangles and polygons, with applications to complex projective toric varieties and open problems.

Contribution

It introduces lattice angle sums, characterizes lattice triangles and polygons, and connects lattice geometry with complex algebraic geometry.

Findings

Established a lattice version of the Euclidean angle sum condition

Derived necessary and sufficient conditions for lattice polygons

Applied results to complex projective toric varieties

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 \pi. 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.