Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.Comment: 21 pages, 25 figure

Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

A generalized Winternitz Theorem

We prove that, for every simple polygon P having k ≥ 1 reflex vertices, there exists a point q ε P such that every half-polygon that contains q contains nearly 1/2(k + 1) times the area of P. We also give a family of examples showing that this result is the best possible

A New Fast and Efficient Conformal Mapping Based Technique for Remote Sensing Data Compression and Transmittal

In this paper we deal with a new technique for large data compression. Contour mapping of two dimensional objects is of fundamental importance in remote sensing and computer vision applications. We present extensive algorithms applied to polygonized, simply-connected contours and reproduce desired shapes using an innovative data compression technique based on conformal mapping. In a previous work3,4, through a conformal mapping process, we demonstrated the ability to 1) recognize shapes, and 2) concisely represent shape boundaries using a set of polynomial coefficients derived in the mapping process. In this work we illustrate how these previous results can be applied to data compression. In particular, in the approach outlined herein, a syntactic representation is formed for polygon shapes whose representation we desire to extract and reproduce compactly. Additionally, we present a problem of concavity in shape boundaries and a proposed solution in which polygons are divided into convex subsets and reconstructed accordingly

Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

Décomposition en Polygones de forme Étoile - Application à la Détection de Pièces

National audienceDans cet article, nous nous intéressons à la décomposition d'une scène polygonale en polygones de forme étoilée. Nous proposons l'emploi d'un algorithme A* afin de définir une partition dont le nombre de régions est minimal. L'approche peut être apparentée à un problème de classification non supervisée des différents segments. En employant le complexe de visibilité, nous sommes ramenés à la manipulation de nombres entiers exclusivement. Nous évaluons les performances de cet algorithme dans le contexte d'un projet de rétroconversion de plans architecturaux afin de détecter les pièces d'un logement dont les murs ont précédemment été extraits

Virtual Glasses Try-on System

Virtual Glasses Try-on System Siyu Quan Recent advances in data-driven modeling have enabled the simulation of wearing the glasses virtually based on 2D images (web-camera). For real-life glasses wearing, it needs not only suitable for appearance, but also comfort. Although the simulations based on 2D images can bring out certain conveniences for customers who want to try on glasses online first, there are still many challenging problems ahead because of the high complexity of simulations for wearing glasses. Obviously, it can hardly tell if the glasses are comfortable or can seat on the customer’s nose correctly. Furthermore, customers may want to take a look from different angle to make sure that the glasses selected are perfect. Such requirements cannot be met by using the simulations based on 2D images so we present an interactive real-time system with simulations for wearing glasses, providing users with a high degree of simulation quality including physics application. With our system the user uses the Kinect sensor or the common web camera as input device to acquire the result of wearing the preferred glasses virtually, which is real-time. Input device captures the user’s face and generate a geometry face-mesh which is expected to be aligned to the face-mesh template we pre-set manually. The 3D data captured from Kinect dramatically improve user’s experience by constructing geometry face mesh

Covering orthogonal polygons with star polygons: The perfect graph approach

AbstractThis paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n3) algorithm