IA Scholar Query: Largest inscribed rectangles in convex polygons.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 20 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Infinitely many virtual geometric triangulations
https://scholar.archive.org/work/peketlkav5favjgikvwjizm5ty
We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.David Futer, Emily Hamilton, Neil R. Hoffmanwork_peketlkav5favjgikvwjizm5tyTue, 20 Sep 2022 00:00:00 GMTHow Curved Creases Enhance the Stiffness and Enable Shape Morphing of Thin-Sheet Structures
https://scholar.archive.org/work/2rwmcqyit5g27nf4h6u3hatddq
When a thin sheet is folded about curved creases, the resulting shape resists loads in some directions and deforms into compact states in other directions. These curved-crease, origami-inspired structures display a number of functional behaviors advantageous to a design, such as tunable stiffness and shape morphing. This dissertation develops an understanding of how these behaviors are related to the crease pattern and how engineers can exploit these traits in their designs through mechanics. The dissertation introduces a method for simulating the structural properties of curved-crease origami using a simplified numerical method called the bar-and-hinge model. Based on the geometry and material of the sheet, stiffness expressions were derived for three deformation behaviors, including stretching of the sheet, bending of the sheet, and folding along the creases. The model is capable of capturing the folding behavior, and the simulated deformed shapes are sufficiently accurate when compared to experiments and to theoretical approximations. This model is used to explore the mechanical characteristics of curved-crease structures throughout the dissertation. Next, the dissertation explores the bending stiffness of curved-crease corrugations that are made by folding thin sheets about curves and without linerboard covers (i.e., flat sheets adhered to the corrugation to give the structure a more isotropic bending stiffness behavior). Curved-creases break symmetry in the corrugation, which allows for a unique property that redistributes stiffness to resist bending deformations in multiple directions. Two formulations for predicting the bending stiffness of any planar-midsurface corrugation were developed and experimentally validated with three-point bending tests. Then, the dissertations explores a unique behavior seen in creased sheets where localized changes in the folding (i.e., pinching of the structure) result in global bending and twisting deformations. It was found that the increase in curvature and torsion of the [...]Steven Woodruff, University, Mywork_2rwmcqyit5g27nf4h6u3hatddqTue, 06 Sep 2022 00:00:00 GMTPosition of the centroid of a planar convex body and the centroid Banach-Mazur distance
https://scholar.archive.org/work/wrluov3lnvej3omk5bj7fdeklu
It is well known that any planar convex body A permits to inscribe an affine-regular hexagon H_A. We prove that the centroid of A belongs to the homothetic image of H_A with ratio 4/21 and the center in the center of H_A.This ratio cannot be enlarged. Moreover, we consider the variant of the Banach-Mazur distance δ_BM^ cen (C, D) of two convex bodies C, D of E^d with the additional requirement that the centroids of them coincide. We prove that δ_BM^ cen (C, D) ≤33/8 for every C, D of E^2.Marek Lassakwork_wrluov3lnvej3omk5bj7fdekluSun, 21 Aug 2022 00:00:00 GMTLargest Inscribed Rectangles in Geometric Convex Sets
https://scholar.archive.org/work/ohu3pvbptbg7ldujjiexc22wye
This paper considers the problem of finding maximum volume (axis-aligned) inscribed boxes in a compact convex set, defined by a finite number of convex inequalities, and presents optimization and geometric approaches for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and squares. To find the largest axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. For 2-dimensional space, a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in convex sets. The optimization approach provides a uniform framework for solving a wide variety of relevant problems. Finally, two computational geometric (1-ε)–approximation algorithms with sub-linear running times are presented that improve the previous results.Mehdi Behrooziwork_ohu3pvbptbg7ldujjiexc22wyeTue, 09 Aug 2022 00:00:00 GMTLinking number and folded ribbon unknots
https://scholar.archive.org/work/gphccrl7u5fpvfmt2bkp7h7s3a
We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3-stick unknots with ribbon linking numbers ±1 and ± 3, and we prove that the minimum folded ribbonlength for n-gons with obtuse interior angles is achieved when the n-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number ± n is bounded from above by 2n.Elizabeth Denne, Troy Larsenwork_gphccrl7u5fpvfmt2bkp7h7s3aFri, 05 Aug 2022 00:00:00 GMTOn Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation
https://scholar.archive.org/work/3inpxcirxvddzbyporoom3h6ia
We first extend the construction of generalized barycentric coordinates (GBC) based on the vertices on the boundary of a polygon Ω to a new kind of GBCs based on vertices inside the Ω of interest. For clarity, the standard GBCs are called boundary GBCs while the new GBCs are called interior GBCs. Then we present an analysis on these two kinds of harmonic GBCs to show that each GBC function whose value is 1 at a vertex (boundary or interior vertex of Ω) decays to zero away from its supporting vertex exponentially fast except for a trivial example. Based on the exponential decay property, we explain how to approximate the harmonic GBC functions locally. That is, due to the locality of these two kinds of GBCs, one can approximate each of these GBC functions by its local versions which is supported over a sub-domain of Ω. The local version of these GBC function will help reduce the computational time for shape deformation in graphical design. Next, with these two kinds of GBC functions at hand, we can use them to approximate the solution of the Dirichlet problem of the Poisson equation. This may provide a more efficient way to solve the Poisson equation by using a computer which has graphical processing unit(GPU) with thousands or more processes than the standard methods using a computer with one or few CPU kernels.Chongyang Deng, Ming-Jun Laiwork_3inpxcirxvddzbyporoom3h6iaSat, 23 Jul 2022 00:00:00 GMTPhysical Interaction and Manipulation of the Environment using Aerial Robots
https://scholar.archive.org/work/df3vacxqnnganly6yohuvoqata
The physical interaction of aerial robots with their environment has countless potential applications and is an emerging area with many open challenges. Fully-actuated multirotors have been introduced to tackle some of these challenges. They provide complete control over position and orientation and eliminate the need for attaching a multi-DoF manipulation arm to the robot. However, there are many open problems before they can be used in real-world applications. Researchers have introduced some methods for physical interaction in limited settings. Their experiments primarily use prototype-level software without an efficient path to integration with real-world applications. We describe a new cost-effective solution for integrating these robots with the existing software and hardware flight systems for real-world applications and expand it to physical interaction applications. On the other hand, the existing control approaches for fully-actuated robots assume conservative limits for the thrusts and moments available to the robot. Using conservative assumptions for these already-inefficient robots makes their interactions even less optimal and may even result in many feasible physical interaction applications becoming infeasible. This work proposes a real-time method for estimating the complete set of instantaneously available forces and moments that robots can use to optimize their physical interaction performance. Finally, many real-world applications where aerial robots can improve the existing manual solutions deal with deformable objects. However, the perception and planning for their manipulation is still challenging. This research explores how aerial physical interaction can be extended to deformable objects. It provides a detection method suitable for manipulating deformable one-dimensional objects and introduces a new perspective on planning the manipulation of these objects.Azarakhsh Keipourwork_df3vacxqnnganly6yohuvoqataWed, 06 Jul 2022 00:00:00 GMTnD-PointCloud Data Management
https://scholar.archive.org/work/uqp6g5hfkzdjxcfzx4ipzb6pgu
In the Geomatics domain, a point cloud refers to a data set that records the coordinates and other attributes of a huge number of points. Conceptually, each of the attributes can be regarded as a dimension to represent a specific type of information, such as time and Level of Importance (LoI). Drastically increasing collection of high dimensional point clouds raises essential demand for smart and highly efficient data management solutions. However, effective tools are missing. File-based solutions require substantial development of data structures and algorithms. Also, with such solutions, enormous effort has to be made to integrate different data types, formats and libraries. By contrast, state-of-the-art DataBase Management Systems (DBMSs) avoid these issues, because they are initially devised for generic use of data. However, DBMSs still present limitations on efficiently indexing non-uniformly distributed points, supporting continuous LoI, and operating high dimensional data. These problems motivate the PhD research which focuses on developing a new DBMS solution. It is aimed at efficiently managing and querying massive nD point clouds to support different types of applications.Haicheng Liuwork_uqp6g5hfkzdjxcfzx4ipzb6pguTue, 28 Jun 2022 00:00:00 GMTContacting Synapse protocol - graphical programming for Icy software
https://scholar.archive.org/work/v54llvtvyzgzxdn26bubsh4nra
This protocol allows to segment synapses contacting GFP labeled cells thanks to wavelet and HK-Means segmentation methods. It will analyze presynaptic bouton's density, shape (roundness), intensity and volume contacting the labeled cells and compare it to the presynaptic boutons away from the cells. It creates an overlay layer of the segmented cells and boutons over the original picture as ROIs (here in blue over the white signal). These ROIs are saved and can be further reused and analyze for many parameters of interest (Intensity, area, perimeter, sphericity,... see below) thanks to ROI statistics bloc (Publication ID: ICY-W5T6J4). The documentation pdf explains how to use this program, and how to install it on Icy software. Icy software being an open software available for PC, Mac or Linux operating system. You will find related picture to test the program on Zenodo (10.5281/zenodo.6756684). The "Contacting synapse.protocol " is the raw protocol that can be used directly in protocol editor within Icy software. Preview of the graphical programming protocol is available by opening the .png file "Contacting synapse.protocol_screenshot.png. All the functional boxes are open and parameters can be changed manually. A more simplier and user friendly version is availbale as "Contacting synapse - user friendly.protocol " (see preview Contacting synapse - user friendly.protocol_screenshot.png) where all the boxes have been closed to let the user access only the dialog boxes.Danglot Lydiawork_v54llvtvyzgzxdn26bubsh4nraSun, 26 Jun 2022 00:00:00 GMTIntrinsic metrics in polygonal domains
https://scholar.archive.org/work/ofpx2wvplfdixoisveq6ic6eya
We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the point to the boundary.D. Dautova, R. Kargar, S. Nasyrov, M. Vuorinenwork_ofpx2wvplfdixoisveq6ic6eyaWed, 08 Jun 2022 00:00:00 GMTPhysical Interaction and Manipulation of the Environment using Aerial Robots
https://scholar.archive.org/work/ufaukj3fozh5vgbgalgyeiif7y
The physical interaction of aerial robots with their environment has countless potential applications and is an emerging area with many open challenges. Fully-actuated multirotors have been introduced to tackle some of these challenges. They provide complete control over position and orientation and eliminate the need for attaching a multi-DoF manipulation arm to the robot. However, there are still several open problems before they can be used in real-world applications. Researchers have introduced some methods for the physical interaction of fully-actuated multirotors in limited settings. Their experiments primarily use prototype-level software without an efficient path to integrating these methods into real-world applications. This thesis describes a new controller design that provides a cost-effective solution for integrating these robots with the existing software and hardware flight systems for real-world applications. It further expands the controller to physical interaction applications to show its flexibility and effectiveness. On the other hand, the existing control approaches for fully-actuated robots assume conservative limits for the thrusts and moments available to the robot. Using conservative assumptions for these already-inefficient robots makes their interactions even less optimal and may even result in many feasible physical interaction applications becoming infeasible. This work proposes a real-time method for estimating the complete set of instantaneously available forces and moments that robots can use to optimize their physical interaction performance. Finally, many real-world applications where aerial robots can improve the existing manual solutions deal with deformable objects. However, the perception of deformable objects and planning for their manipulation is still challenging. Additionally, no studies have been performed to analyze the requirements of aerial tasks that involve deformable objects. This research explores how aerial physical interaction can be extended to deformable object [...]Azarakhsh Keipourwork_ufaukj3fozh5vgbgalgyeiif7yMon, 06 Jun 2022 00:00:00 GMTLarge k-gons in a 1.5D Terrain
https://scholar.archive.org/work/7iar664l6nghhjh3vt7zzelbcm
Given is a 1.5D terrain 𝒯, i.e., an x-monotone polygonal chain in ℝ^2. For a given 2≤ k≤ n, our objective is to approximate the largest area or perimeter convex polygon of exactly or at most k vertices inside 𝒯. For a constant k>3, we design an FPTAS that efficiently approximates the largest convex polygons with at most k vertices, within a factor (1-ϵ). For the case where k=2, we design an O(n) time exact algorithm for computing the longest line segment in 𝒯, and for k=3, we design an O(n log n) time exact algorithm for computing the largest-perimeter triangle that lies within 𝒯.Vahideh Keikhawork_7iar664l6nghhjh3vt7zzelbcmMon, 06 Jun 2022 00:00:00 GMTReshaping Convex Polyhedra
https://scholar.archive.org/work/zinzcqgcxrdkbpwinzteewl764
Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P, and to continuously folding P onto Q. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.Joseph O'Rourke, Costin Vilcuwork_zinzcqgcxrdkbpwinzteewl764Sat, 21 May 2022 00:00:00 GMTMusic Encoding Conference Proceedings 2021. 19–22 July, 2021 University of Alicante (Spain): Onsite & Online. Edited by Stefan Münnich and David Rizo
https://scholar.archive.org/work/khnwkuihvzcalchduxno6dnv7u
Conference proceedings of the Music Encoding Conference 2021 with Foreword by Stefan Münnich and David Rizo.HC User, Stefan Münnich, David Rizowork_khnwkuihvzcalchduxno6dnv7uWed, 18 May 2022 00:00:00 GMTTowards a Geometric Understanding of the 4-Dimensional Point Groups
https://scholar.archive.org/work/3r2lvvgst5g4bjij4unz2fqhum
We classify the finite groups of orthogonal transformations in 4-space, and we study these groups from the viewpoint of their geometric action, using polar orbit polytopes. For one type of groups (the toroidal groups), we develop a new classification based on their action on an invariant torus, while we rely on classic results for the remaining groups. As a tool, we develop a convenient parameterization of the oriented great circles on the 3-sphere, which leads to (oriented) Hopf fibrations in a natural way.Laith Rastanawi, Günter Rotework_3r2lvvgst5g4bjij4unz2fqhumTue, 10 May 2022 00:00:00 GMTApproximate Convex Decomposition for 3D Meshes with Collision-Aware Concavity and Tree Search
https://scholar.archive.org/work/eqgmnhknj5eadlqjsgwnmkluxq
Approximate convex decomposition aims to decompose a 3D shape into a set of almost convex components, whose convex hulls can then be used to represent the input shape. It thus enables efficient geometry processing algorithms specifically designed for convex shapes and has been widely used in game engines, physics simulations, and animation. While prior works can capture the global structure of input shapes, they may fail to preserve fine-grained details (e.g., filling a toaster's slots), which are critical for retaining the functionality of objects in interactive environments. In this paper, we propose a novel method that addresses the limitations of existing approaches from three perspectives: (a) We introduce a novel collision-aware concavity metric that examines the distance between a shape and its convex hull from both the boundary and the interior. The proposed concavity preserves collision conditions and is more robust to detect various approximation errors. (b) We decompose shapes by directly cutting meshes with 3D planes. It ensures generated convex hulls are intersection-free and avoids voxelization errors. (c) Instead of using a one-step greedy strategy, we propose employing a multi-step tree search to determine the cutting planes, which leads to a globally better solution and avoids unnecessary cuttings. Through extensive evaluation on a large-scale articulated object dataset, we show that our method generates decompositions closer to the original shape with fewer components. It thus supports delicate and efficient object interaction in downstream applications. We will release our implementation to facilitate future research.Xinyue Wei, Minghua Liu, Zhan Ling, Hao Suwork_eqgmnhknj5eadlqjsgwnmkluxqThu, 05 May 2022 00:00:00 GMTA Sequential MPC Approach to Reactive Planning for Bipedal Robots
https://scholar.archive.org/work/bffrzesl55gibmrabqsl4lw34i
This paper presents a sequential Model Predictive Control (MPC) approach to reactive motion planning for bipedal robots in dynamic environments. The approach relies on a sequential polytopic decomposition of the free space, which provides an ordered collection of mutually intersecting obstacle free polytopes and waypoints. These are subsequently used to define a corresponding sequence of MPC programs that drive the system to a goal location avoiding static and moving obstacles. This way, the planner focuses on the free space in the vicinity of the robot, thus alleviating the need to consider all the obstacles simultaneously and reducing computational time. We verify the efficacy of our approach in high-fidelity simulations with the bipedal robot Digit, demonstrating robust reactive planning in the presence of static and moving obstacles.Kunal Sanjay Narkhede, Abhijeet Mangesh Kulkarni, Dhruv Ashwinkumar Thanki, Ioannis Poulakakiswork_bffrzesl55gibmrabqsl4lw34iSat, 30 Apr 2022 00:00:00 GMTn-gon centers and central lines
https://scholar.archive.org/work/kzzwpgho7bhzpfafj232t23tcm
In this paper we provide a review of the concept of center of a n-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for n-gons for n≥ 3 and establish its basic properties.Marta Farré Puiggalí, Luis Felipe Prieto-Martínezwork_kzzwpgho7bhzpfafj232t23tcmFri, 15 Apr 2022 00:00:00 GMTConvex geometry
https://scholar.archive.org/work/psttss32trahnda6v3h5tif7te
ACM 204, Winter 2019Joel A. Tropp, Caltech CMS Lecture Noteswork_psttss32trahnda6v3h5tif7teWed, 13 Apr 2022 00:00:00 GMTLearning Mixed-Integer Convex Optimization Strategies for Robot Planning and Control
https://scholar.archive.org/work/hlmvhcvyqncl5bhst5ns4v2rbe
Mixed-integer convex programming (MICP) has seen significant algorithmic and hardware improvements with several orders of magnitude solve time speedups compared to 25 years ago. Despite these advances, MICP has been rarely applied to real-world robotic control because the solution times are still too slow for online applications. In this work, we present the CoCo (Combinatorial Offline, Convex Online) framework to solve MICPs arising in robotics at very high speed. CoCo encodes the combinatorial part of the optimal solution into a strategy. Using data collected from offline problem solutions, we train a multiclass classifier to predict the optimal strategy given problem-specific parameters such as states or obstacles. Compared to previous approaches, we use task-specific strategies and prune redundant ones to significantly reduce the number of classes the predictor has to select from, thereby greatly improving scalability. Given the predicted strategy, the control task becomes a small convex optimization problem that we can solve in milliseconds. Numerical experiments on a cart-pole system with walls, a free-flying space robot, and task-oriented grasps show that our method provides not only 1 to 2 orders of magnitude speedups compared to state-of-the-art solvers but also performance close to the globally optimal MICP solution.A. Cauligi, P. Culbertson, B. Stellato, D. Bertsimas, M. Schwager, M. Pavonework_hlmvhcvyqncl5bhst5ns4v2rbeMon, 11 Apr 2022 00:00:00 GMT