Most haptic environments are based on single point interactions whereas in practice, object manipulation requires multiple contact points between the object, fingers, thumb and palm. The Friction Cone Algorithm was developed specifically to work well in a multi-finger haptic environment where object manipulation would occur. However, the Friction Cone Algorithm has two shortcomings when applied...

The generation of random simple polygon and the pseudo-triangulation of a polygon are regarded as the proposed problems in computational geometry. The production of a random polygon is used in the context of the consideration of the accuracy of algorithms. In this paper, a new algorithm is presented to generate a simple spiral polygon on a set of random points S in the plane using convex hull l...

We propose new measures to evaluate to which extent the shape of a given convex polygon is close to the shape of some regular polygon. We prove that our parameters satisfy several reasonable requirements and provide algorithms for their efficient computation. The properties we mostly focus on are the facts that regular polygons are equilateral, equiangular and have radial symmetry.

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...

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.