1,870 research outputs found
Separation-Sensitive Collision Detection for Convex Objects
We develop a class of new kinetic data structures for collision detection
between moving convex polytopes; the performance of these structures is
sensitive to the separation of the polytopes during their motion. For two
convex polygons in the plane, let be the maximum diameter of the polygons,
and let be the minimum distance between them during their motion. Our
separation certificate changes times when the relative motion of
the two polygons is a translation along a straight line or convex curve,
for translation along an algebraic trajectory, and for
algebraic rigid motion (translation and rotation). Each certificate update is
performed in time. Variants of these data structures are also
shown that exhibit \emph{hysteresis}---after a separation certificate fails,
the new certificate cannot fail again until the objects have moved by some
constant fraction of their current separation. We can then bound the number of
events by the combinatorial size of a certain cover of the motion path by
balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM
Symposium on Discrete Algorithms, 1999; see also
http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission
with camera-ready versio
Artisan procedures to generate uniform tilings
This paper try to prove how artisans c
ould discover all uniform tilings and very
interesting others us
ing artisanal combinatorial pro
cedures without
having to use
mathematical procedures out of their reac
h. Plane Geometry started up his way through
History by means of fundamental
drawing tools: ruler and co
mpass. Artisans used same
tools to carry out their orna
mental patterns but at some
point they began to work
manually using physical representations of fi
gures or tiles previously drawing by means
of ruler and compass. That is an important step for craftsman because this way provides
tools that let him come in
the world of symmetry opera
tions and empirical knowledge
of symmetry groups. Artisans started up to pr
oduce little wooden, ceramic or clay tiles
and began to experiment with them by means of
joining pieces whether edge to edge or
vertex to vertex in that way so it can c
over the plane without gaps. Economy in making
floor or ceramic tiles could be most important reason to develop these procedures. This
empiric way to develop tilings led not only
to discover all uniform tilings but later
discovering of aperiodic tilings
IST Austria Thesis
In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Discretization of Planar Geometric Cover Problems
We consider discretization of the 'geometric cover problem' in the plane:
Given a set of points in the plane and a compact planar object ,
find a minimum cardinality collection of planar translates of such that
the union of the translates in the collection contains all the points in .
We show that the geometric cover problem can be converted to a form of the
geometric set cover, which has a given finite-size collection of translates
rather than the infinite continuous solution space of the former. We propose a
reduced finite solution space that consists of distinct canonical translates
and present polynomial algorithms to find the reduce solution space for disks,
convex/non-convex polygons (including holes), and planar objects consisting of
finite Jordan curves.Comment: 16 pages, 5 figure
Sampling-based coverage path planning for complex 3D structures
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 173-186).Path planning is an essential capability for autonomous robots, and many applications impose challenging constraints alongside the standard requirement of obstacle avoidance. Coverage planning is one such task, in which a single robot must sweep its end effector over the entirety of a known workspace. For two-dimensional environments, optimal algorithms are documented and well-understood. For threedimensional structures, however, few of the available heuristics succeed over occluded regions and low-clearance areas. This thesis makes several contributions to sampling-based coverage path planning, for use on complex three-dimensional structures. First, we introduce a new algorithm for planning feasible coverage paths. It is more computationally efficient in problems of complex geometry than the well-known dual sampling method, especially when high-quality solutions are desired. Second, we present an improvement procedure that iteratively shortens and smooths a feasible coverage path; robot configurations are adjusted without violating any coverage constraints. Third, we propose a modular algorithm that allows the simple components of a structure to be covered using planar, back-and-forth sweep paths. An analysis of probabilistic completeness, the first of its kind applied to coverage planning, accompanies each of these algorithms, as well as ensemble computational results. The motivating application throughout this work has been autonomous, in-water ship hull inspection. Shafts, propellers, and control surfaces protrude from a ship hull and pose a challenging coverage problem at the stern. Deployment of a sonar-equipped underwater robot on six large vessels has led to robust operations that yield triangle mesh models of these structures; these models form the basis for planning inspections at close range. We give results from a coverage plan executed at the stern of a US Coast Guard Cutter, and results are also presented from an indoor experiment using a precision scanning laser and gantry positioning system.by Brendan J. Englot.Ph.D
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