1,199 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
Kinetic collision detection between two simple polygons
AbstractWe design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing
Motivated by the desire to cope with data imprecision, we study methods for
taking advantage of preliminary information about point sets in order to speed
up the computation of certain structures associated with them.
In particular, we study the following problem: given a set L of n lines in
the plane, we wish to preprocess L such that later, upon receiving a set P of n
points, each of which lies on a distinct line of L, we can construct the convex
hull of P efficiently. We show that in quadratic time and space it is possible
to construct a data structure on L that enables us to compute the convex hull
of any such point set P in O(n alpha(n) log* n) expected time. If we further
assume that the points are "oblivious" with respect to the data structure, the
running time improves to O(n alpha(n)). The analysis applies almost verbatim
when L is a set of line-segments, and yields similar asymptotic bounds. We
present several extensions, including a trade-off between space and query time
and an output-sensitive algorithm. We also study the "dual problem" where we
show how to efficiently compute the (<= k)-level of n lines in the plane, each
of which lies on a distinct point (given in advance).
We complement our results by Omega(n log n) lower bounds under the algebraic
computation tree model for several related problems, including sorting a set of
points (according to, say, their x-order), each of which lies on a given line
known in advance. Therefore, the convex hull problem under our setting is
easier than sorting, contrary to the "standard" convex hull and sorting
problems, in which the two problems require Theta(n log n) steps in the worst
case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at
SoCG 201
New Geometric Data Structures for Collision Detection
We present new geometric data structures for collision detection and more, including: Inner Sphere Trees - the first data structure to compute the peneration volume efficiently. Protosphere - an new algorithm to compute space filling sphere packings for arbitrary objects. Kinetic AABBs - a bounding volume hierarchy that is optimal in the number of updates when the objects deform. Kinetic Separation-List - an algorithm that is able to perform continuous collision detection for complex deformable objects in real-time. Moreover, we present applications of these new approaches to hand animation, real-time collision avoidance in dynamic environments for robots and haptic rendering, including a user study that exploits the influence of the degrees of freedom in complex haptic interactions. Last but not least, we present a new benchmarking suite for both, peformance and quality benchmarks, and a theoretic analysis of the running-time of bounding volume-based collision detection algorithms
Self-assembly in polyoxometalate and metal coordination-based systems: synthetic approaches and developments
Utilizing new experimental approaches and gradual understanding of the underlying chemical processes has led to advances in the self-assembly of inorganic and metalâorganic compounds at a very fast pace over the last decades. Exploitation of unveiled information originating from initial experimental observations has sparked the development of new families of compounds with unique structural characteristics and functionalities. The main source of inspiration for numerous research groups originated from the implementation of the design element along with the discovery of new chemical components which can self-assemble into complex structures with wide range of sizes, topologies and functionalities. Not only do self-assembled inorganic and metalâorganic chemical systems belong to families of compounds with configurable structures, but also have a vast array of physical properties which reflect the chemical information stored in the various âmodularâ molecular subunits. The purpose of this short review article is not the exhaustive discussion of the broad field of inorganic and metalâorganic chemical systems, but the discussion of some representative examples from each category which demonstrate the implementation of new synthetic approaches and design principles
Kinetic collision detection for balls rolling on a plane
This abstract presents a first step towards kinetic col- lision detection in 3 dimensions. In particular, we design a compact and responsive kinetic data struc- ture (KDS) for detecting collisions between n balls of arbitrary sizes rolling on a plane. The KDS has size O(n log n) and can handle events in O(log n) time. The structure processes O(n2) events in the worst case, assuming that the objects follow low-degree al- gebraic trajectories. The full paper [1] presents ad- ditional results for convex fat 3-dimensional objects that are free-flying in R3
Algorithms for fat objects : decompositions and applications
Computational geometry is the branch of theoretical computer science that deals with algorithms and data structures for geometric objects. The most basic geometric objects include points, lines, polygons, and polyhedra. Computational geometry has applications in many areas of computer science, including computer graphics, robotics, and geographic information systems. In many computational-geometry problems, the theoretical worst case is achieved by input that is in some way "unrealistic". This causes situations where the theoretical running time is not a good predictor of the running time in practice. In addition, algorithms must also be designed with the worst-case examples in mind, which causes them to be needlessly complicated. In recent years, realistic input models have been proposed in an attempt to deal with this problem. The usual form such solutions take is to limit some geometric property of the input to a constant. We examine a specific realistic input model in this thesis: the model where objects are restricted to be fat. Intuitively, objects that are more like a ball are more fat, and objects that are more like a long pole are less fat. We look at fat objects in the context of five different problemsâtwo related to decompositions of input objects and three problems suggested by computer graphics. Decompositions of geometric objects are important because they are often used as a preliminary step in other algorithms, since many algorithms can only handle geometric objects that are convex and preferably of low complexity. The two main issues in developing decomposition algorithms are to keep the number of pieces produced by the decomposition small and to compute the decomposition quickly. The main question we address is the following: is it possible to obtain better decompositions for fat objects than for general objects, and/or is it possible to obtain decompositions quickly? These questions are also interesting because most research into fat objects has concerned objects that are convex. We begin by triangulating fat polygons. The problem of triangulating polygonsâthat is, partitioning them into triangles without adding any verticesâhas been solved already, but the only linear-time algorithm is so complicated that it has never been implemented. We propose two algorithms for triangulating fat polygons in linear time that are much simpler. They make use of the observation that a small set of guards placed at points inside a (certain type of) fat polygon is sufficient to see the boundary of such a polygon. We then look at decompositions of fat polyhedra in three dimensions. We show that polyhedra can be decomposed into a linear number of convex pieces if certain fatness restrictions aremet. We also show that if these restrictions are notmet, a quadratic number of pieces may be needed. We also show that if we wish the output to be fat and convex, the restrictions must be much tighter. We then study three computational-geometry problems inspired by computer graphics. First, we study ray-shooting amidst fat objects from two perspectives. This is the problem of preprocessing data into a data structure that can answer which object is first hit by a query ray in a given direction from a given point. We present a new data structure for answering vertical ray-shooting queriesâthat is, queries where the rayâs direction is fixedâas well as a data structure for answering ray-shooting queries for rays with arbitrary direction. Both structures improve the best known results on these problems. Another problem that is studied in the field of computer graphics is the depth-order problem. We study it in the context of computational geometry. This is the problem of finding an ordering of the objects in the scene from "top" to "bottom", where one object is above the other if they share a point in the projection to the xy-plane and the first object has a higher z-value at that point. We give an algorithm for finding the depth order of a group of fat objects and an algorithm for verifying if a depth order of a group of fat objects is correct. The latter algorithm is useful because the former can return an incorrect order if the objects do not have a depth order (this can happen if the above/below relationship has a cycle in it). The first algorithm improves on the results previously known for fat objects; the second is the first algorithm for verifying depth orders of fat objects. The final problem that we study is the hidden-surface removal problem. In this problem, we wish to find and report the visible portions of a scene from a given viewpointâthis is called the visibility map. The main difficulty in this problem is to find an algorithm whose running time depends in part on the complexity of the output. For example, if all but one of the objects in the input scene are hidden behind one large object, then our algorithm should have a faster running time than if all of the objects are visible and have borders that overlap. We give such an algorithm that improves on the running time of previous algorithms for fat objects. Furthermore, our algorithm is able to handle curved objects and situations where the objects do not have a depth orderâtwo features missing from most other algorithms that perform hidden surface removal
Kinetic and dynamic data structures for convex hulls and upper envelopes
AbstractLet S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O(n2ÎČs+2(n)logn) critical events, each in O(log2n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, ÎČs(q)=λs(q)/q, and λs(q) is the maximum length of DavenportâSchinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1â28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Recommended from our members
Investigations of Surface-Tension Effects Due to Small-Scale Complex Boundaries
The earliest man-made irrigation systems in recorded history date back to the ancient Egypt and Mesopotamia era. After thousands of years of experience, exploration, and experimenting, mankind have learned how to construct canals and dams and use pipes and pumps to direct and control water flow, but till this day, there are still some behaviors of water and other simple fluids that surprise us. One such example is the lotus effect: a surface-tension effect which allows raindrops to roll freely on a lotus leaf as if they were drops of mercury. One of the key factors that determine how a fluid system behave is the size-scale. Fluids flow at small scales very differently than they do at large scales. The standard comparing to which small and large are defined is the capillary length. A number of surface-tension related phenomena are unfamiliar because they are only noticeable at length-scales of a few millimeters or below, and they look nothing like what we would expect fluids to behave when dominated by gravity. As fascinating as many of them may seem at first glance, surface-tension phenomena are actually not that far away from our daily lives.
Surface tension is everywhere because it costs energy to create areas of surfaces and interfaces, just like it costs energy to deform a solid (resulting in elasticity) or to elevate a weight (resulting in gravity). To minimize energy, a surface or an interface has the tendency to contract, and this tendency generates surface tension. The size of a system significantly affects the relative strengths of surface-tension effects comparing to effects of body forces, most commonly gravity. By equating the estimated magnitudes of surface tension and gravitational forces of a system, a length scale, know as the capillary length, can be defined. The capillary length of water on earth is about 2.7 mm. At the length scale of everyday objects, which is usually above the capillary length, surface-tension effects are not always prominent, because at those scales the competing force, gravity, is often much stronger. That is why the surface of a glass of water is more or less flat. However, as the size-scale decreases, surface tension decreases a lot slower than gravity, so when the size of a fluid system gets down to below the capillary length, surface tension takes over.
One of the defining characteristics of this moment in human history, is the tremendous efforts we are putting into the research and engineering of micro- and nano-scale materials and structures â systems where surface tension is often the predominant force. It is important to study surface-tension effects so that we can use them to our advantage. In this Ph.D. dissertation, we have investigated some important surface-tension phenomena including capillarity, wetting, and wicking. We mainly focus on the geometric aspects of these problems, and to learn about how structures affect properties. Understanding these phenomena can help develop fabrication methods (Chapter 2), study surface properties (Chapter 3), and design useful devices (Chapter 4) at scales below the capillary length.
In the first project (Chapter 2), we used numerical simulations and experiments to study the meniscus of a fluid confined in capillaries with complicated cross-sectional geometries. In the simulations, we computed the three-dimensional shapes of the menisci formed in polygonal and star-shaped capillaries with sharp or rounded corners. Height variations across the menisci were used to quantify the effect of surface tension. Analytical solutions were derived for all the cases where the cross-sectional geometry was a regular polygon or a regular star-shape. Power indices that characterize the effects of corner rounding were extracted from simulation results. These findings can serve as guide for fabrications of unconventional three-dimensional structures in Capillary Force Lithography experiments [J. Feng (2011) (a)]. Experimental demonstrations of the working principle was also performed. Although quantitative matching between simulation and experimental results was not achieved due to the limitation of material properties, clear qualitative trends were observed and interesting three-dimensional nano-structures were produced.
A second project (Chapter 3) focused on developing techniques to produce three-dimensional hierarchically structured superhydrophobic surfaces with high aspect ratios. We experimented with two different high-throughput electron-beam-lithography processes featuring single and dual electron-beam exposures. After a surface modification procedure with a hydrophobic silane, the structured surfaces exhibited two distinct superhydrophobic behaviors â high and low adhesion. While both types of superhydrophobic surfaces exhibited very high (approximately 160_) water advancing contact angles, the water receding contact angles on these two different types of surfaces differed by about 50_ _ 60_, with the low-adhesion surfaces at about 120_ _ 130_ and the high-adhesion surfaces at about 70_ _ 80_. Characterizations of both the microscopic structures and macroscopic wetting properties of these product surfaces allowed us to pinpoint the structural features responsible for specific wetting properties. It is found that the advancing contact angle was mainly determined by the primary structures while the receding contact angle is largely affected by the side-wall slope of the secondary features. This study established a platform for further exploration of the structure aspects of surface wettability [J. Feng (2011) (b)].
In the third and final project (Chapter 4), we demonstrated a new type of microfluidic channel that enable asymmetric wicking of wetting fluids based on structure-induced direction-dependent surface-tension effect. By decorating the side-walls of open microfluidic channels with tilted fins, we were able to experimentally demonstrate preferential wicking behaviors of various IPA-water mixtures with a range of contact angles in these channels. A simplified 2D model was established to explain the wicking asymmetry, and a complete 3D model was developed to provide more accurate quantitative predictions. The design principles developed in this study provide an additional scheme for controlling the spreading of fluids [J. Feng (2012)].
The research presented in this dissertation spreads out across a wide range of physical phenomena (wicking, wetting, and capillarity), and involves a number of computational and experimental techniques, yet all of these projects are intrinsically united under a common theme: we want to better understand how simple fluids respond to small-scale complex surface structures as manifestations of surface-tension effects. We hope our findings can serve as building blocks for a larger scale endeavor of scientific research and engineering development. After all, the pursue of knowledge is most meaningful if the results improve the well-being of the society and the advancement of humanity
- âŠ