2,916 research outputs found
Elementary notions of lattice trigonometry
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.Comment: 49 pages; 16 figure
The cross covariogram of a pair of polygons determines both polygons, with a few exceptions
The cross covariogram g_{K,L} of two convex sets K and L in R^n is the
function which associates to each x in R^n the volume of the intersection of K
and L+x.
Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture
on the covariogram problem, that asserts that any planar convex body K is
determined by the knowledge of g_{K,K}.
The problem of determining the sets from their covariogram is relevant in
probability, in statistical shape recognition and in the determination of the
atomic structure of a quasicrystal from X-ray diffraction images.
We prove that when K and L are convex polygons (and also when K and L are
planar convex cones) g_{K,L} determines both K and L, up to a described family
of exceptions. These results imply that, when K and L are in these classes, the
information provided by the cross covariogram is so rich as to determine not
only one unknown body, as required by Matheron's conjecture, but two bodies,
with a few classified exceptions.
These results are also used by Bianchi [Bia] to prove that any convex
polytope P in R^3 is determined by g_{P,P}.Comment: 26 pages, 9 figure
Introducing symplectic billiards
In this article we introduce a simple dynamical system called symplectic
billiards. As opposed to usual/Birkhoff billiards, where length is the
generating function, for symplectic billiards symplectic area is the generating
function. We explore basic properties and exhibit several similarities, but
also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure
- …