Given a convex polygon P in the projective plane we can form a finite “grid” of points by taking the pairwise intersections of the lines extending the edges of P . When P is a Poncelet polygon we show that this grid is contained in a finite union of ellipses and hyperbolas and derive other related geometric information about the grid.

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap each other. We present an approximation scheme for this problem assuming that the container is a convex polygon. O...

The situation is somewhat different when the aligned rectangles are replaced by similar copies of a given convex polygon. More precisely, suppose that 0 is a distribution of N points in the unit square U ̄ [0, 1]#, treated as a torus. Suppose that AXU is a closed convex polygon of diameter less than 1 and with centre of gravity at the origin O. For every real number r satisfying 0% r% 1 and ever...

reliability investigation has always been one of the most important issues in power systems planning. the outages rate in power system reflects the fact that more attentions should be paid on reliability indices to supply consumers with uninterrupted power. using reliability indices in economic dispatch problem may lead to the system load demand with high reliability and low probability of powe...

This paper presents a GPU (Graphics Processing Units) implementation of dynamic programming for the optimal polygon triangulation. Recently, GPUs can be used for general purpose parallel computation. Users can develop parallel programs running on GPUs using programming architecture called CUDA (Compute Unified Device Architecture) provided by NVIDIA. The optimal polygon triangulation problem fo...

A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in [3]. For linear equations in a convex polygon [2,4,5], this method: (a) Expresses the solution q(x,y) in the form of an integral (generalized inverse Fourier transform) in the complex k-plane involving a certain function q̂(k) (generaliz...

We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then

The billiard in a polygon is not always ergodic and never K-mixing or Bernoulli. Here we consider billiard tables by attaching disks to each vertex of an arbitrary simply connected, convex polygon. We show that the billiard on such a table is ergodic, K-mixing and Bernoulli.

We consider the problem of nding a strip separating between two polygons, whose intersection with a third (convex) polygon is of maximum area. We present an optimal linear-time algorithm for computing the optimum strip. When the third polygon is not convex, the running time of the algorithm is quadratic in the size of the input. The application in mind is the piecewise-linear surface interpolat...

We derive a formula for an n-th order divided difference of the inverse of a function. The formula has a simple and surprising structure: it is a sum over partitions of a convex polygon with n+1 vertices. The formula provides a numerically stable method of computing divided differences of k-th roots. It also provides a new way of enumerating all partitions of a convex polygon of a certain type,...