We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout w...

Let S be a finite set of n points in the plane in general position. A k-hole of S is a simple polygon with k vertices from S and no points of S in its interior. A simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. Moreover, a point set S is l-convex if there exists an l-convex polygonalization of S. Considering a typical Erdős-Szek...

We present a new system for robustly performing Boolean operations on linear, 3D polyhedra. Our system is exact, meaning that all internal numeric predicates are exactly decided in the sense of exact geometric computation. Our BSP-tree based system is 16-28× faster at performing iterative computations than CGAL’s Nef Polyhedra based system, the current best practice in robust Boolean operations...

Let Tn denote the set of triangulations of a convex polygon K with n sides We study the random walk on Tn whose transitions are ips of one of the n internal diagonals of the current triangulation the choice of diagonal being random By bounding the conductance of this graph we show that the walk mixes rapidly namely in time O n A direct argument is given for the fact that the mixing rate is at l...

Let S = {l1, l2, l3, . . . , ln} be a set of n vertical line segments in the plane. Though not essential, to simplify proofs we assume that no two li’s are on the same vertical line. A convex polygon weakly intersects S if it contains a point of each line segment on its boundary or interior. In this paper, we propose an O(n log n) algorithm for the problem of finding a minimum area convex polyg...

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m + n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest po...

This paper considers reconngurations of polygons, where each polygon edge is a rigid link, no two of which can cross during the motion. We prove that one can reconngure any monotone polygon into a convex polygon; a polygon is monotone if any vertical line intersects the interior at a (possibly empty) interval. Our algorithm computes in O(n 2) time a sequence of O(n 2) moves, each of which rotat...

Given a convex polygon P with n vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover P . We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius r satisfying r ≤ 2ropt using O(1) space, where ropt is the optimum solution. Nex...

Dimer models are introduced by string theorists (see e.g. [11, 12, 13, 16, 17, 18]) to study supersymmetric quiver gauge theories in four dimensions. A dimer model is a bicolored graph on a 2-torus encoding the information of a quiver with relations. If a dimer model is non-degenerate, then the moduli space Mθ of stable representations of the quiver with dimension vector (1, . . . , 1) with res...

Let P be a set of n points on the plane in general position. We say that a set Γ of convex polygons with vertices in P is a convex decomposition of P if: Union of all elements in Γ is the convex hull of P, every element in Γ is empty, and for any two different elements of Γ their interiors are disjoint. A minimal convex decomposition of P is a convex decomposition Γ′ such that for any two adjac...