Elementary notions of lattice trigonometry.

*(English)*Zbl 1155.11035Given a full-rank lattice in a two-dimensional oriented vector space, the notation lattice polygon is used to denote a polygon whose vertices are points of this lattice. Then the angles of any lattice triangle are called lattice angles.

In the paper under review, the author defines and studies lattice trigonometric functions of the lattice angles. These functions are invariant under the action of the group of affine transformations preserving the lattice, analogous to the invariance of the usual trigonometric functions under the action of the group of Euclidean length-preserving transformations of a Euclidean space. One of the main results of the paper is a classification of lattice triangles up to the lattice-affine equivalence relation, which is a step toward the general classification problem of convex lattice polygons. The lattice triangles are described in terms of lattice sums of lattice angles. In particular, the author presents a sum formula for the lattice tangents of ordinary lattice angles of lattice triangles, which is a generalization of the Euclidean statement: three angles are the angles of some triangle iff their sum equals \(\pi\). The author gives a geometrical interpretation of lattice tangents in terms of ordinary continued fractions, and then generalizes this interpretation to extended lattice angles in terms of sails in the sense of Klein.

Beyond the study of lattice triangles, the author is also able to give a necessary and sufficient condition for an ordered \(n\)-tuple of angles to be angles of a convex lattice polygon. In the appendices to the paper, the author also discusses some applications of this theory to the study of complex projective toric varieties, and formulates criteria for lattice congruence of lattice triangles.

In the paper under review, the author defines and studies lattice trigonometric functions of the lattice angles. These functions are invariant under the action of the group of affine transformations preserving the lattice, analogous to the invariance of the usual trigonometric functions under the action of the group of Euclidean length-preserving transformations of a Euclidean space. One of the main results of the paper is a classification of lattice triangles up to the lattice-affine equivalence relation, which is a step toward the general classification problem of convex lattice polygons. The lattice triangles are described in terms of lattice sums of lattice angles. In particular, the author presents a sum formula for the lattice tangents of ordinary lattice angles of lattice triangles, which is a generalization of the Euclidean statement: three angles are the angles of some triangle iff their sum equals \(\pi\). The author gives a geometrical interpretation of lattice tangents in terms of ordinary continued fractions, and then generalizes this interpretation to extended lattice angles in terms of sails in the sense of Klein.

Beyond the study of lattice triangles, the author is also able to give a necessary and sufficient condition for an ordered \(n\)-tuple of angles to be angles of a convex lattice polygon. In the appendices to the paper, the author also discusses some applications of this theory to the study of complex projective toric varieties, and formulates criteria for lattice congruence of lattice triangles.

Reviewer: Lenny Fukshansky (Claremont)

##### MSC:

11J70 | Continued fractions and generalizations |

11A55 | Continued fractions |

52C99 | Discrete geometry |