In this note we analyze the topology of the moduli spaces of configurations in the euclidian space R of all linearly immersed polygonal circles with either fixed lengths for the sides or one side allowed to vary. Specifically, this means that the allowed maps of a k-gon 〈l1, l2, . . . , lk〉 where the li are the lengths of the successive sides, are specified by an ordered k-tuple of points in R,...

The Erdős–Szekeres theorem inspired a lot of research. A frequent topic in this area is the study of the existence of so-called empty convex polygons in finite planar point sets. Let P be a finite set of points in general position in the plane. A convex k-gon G is called a k-hole (or empty convex k-gon) of P , if all vertices of G lie in P and no point of P lies inside G. Frequently we will mea...

SIM WON LEE, JAE HWAN KIM, MAN GYUN NA, DONG SU KIM, KEUK JONG YU, and HAN GON KIM Department of Nuclear Engineering, Chosun University 375 Seosuk-dong, Dong-gu, Gwangju 501-759, Republic of Korea Korea Atomic Energy Research Institute Dukjin-dong, Yuseong-gu, Daejon 305-600, Republic of Korea Central Research Institute, KHNP 25-1 jang-dong, Deajeon 305-343, Republic of Korea Corresponding auth...

Suppose ↵, are concentric. Let a and b denote the radii of ↵ and respectively, and assume a > b. Clearly, such an arrangement of circles i exists only when the centers C1, . . . , Cn of the n circles 1, . . . n are the vertices of a regular n-gon, whose center is the common center O of ↵ and . Consider then a point of tangency T between two circles i and i+1, and the triangle 4OCiT . The edge O...

Let G he a graph . We say that G contains a complete k-gon if there are !c vertices of G any two of which are connected by an edge, we say that it contains a complete topological k-gon if it contains k vertices any two of which are connected by paths no two of which have a common vertex (except endpoints) . Following G . DIRAC we will denote complete k-gons by -k = and complete topological k-go...

In this paper, we examine Ramsey’s theorem, originally a combinatorial result, and use it to prove a result of a geometric nature, the ErdősSzekeres theorem on convex polygons: given any positive integer k, it is possible to find a least positive integer, ES(k), such that any set of at least ES(k) points is guaranteed to contain the vertex set of a convex k-gon. We generalize the problem to hig...

We need some definitions from [1]. Let C ⊂ R2 be a convex body. A chord pq of C is called an affine diameter of C, if there is no longer parallel chord in C. The ratio of |ab| to 1 2 |a′b′|, where a′b′ is an affine diameter of C parallel to ab, is called the C-length of ab, or the relative length of ab, if there is no doubt about C. We denote it by λC(ab). Denote by λn the relative length of a ...

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the str...

Given a polytope P ⊂ R, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone S+. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P . One of the simplest families of polytopes with interesting symmetries are regul...

Yan-Bin Jia Michael Erdmann The Robotics Institute School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 Abstract Industrial assembly involves sensing the pose (orientation and position) of a part. E cient and reliable sensing strategies can be developed for an assembly task if the shape of the part is known in advance. In this paper we investigate the problem of deter...