Elementary notions of lattice trigonometry
Combinatorics
2009-06-07 v3 Number Theory
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.
Keywords
Cite
@article{arxiv.math/0604129,
title = {Elementary notions of lattice trigonometry},
author = {Oleg Karpenkov},
journal= {arXiv preprint arXiv:math/0604129},
year = {2009}
}
Comments
49 pages; 16 figures