Showing papers on "Computational geometry published in 1990"

Determining the Separation of Preprocessed Polyhedra - A Unified Approach

Abstract: We show how (now familiar) hierarchical representations of (convex) polyhedra can be used to answer various separation queries efficiently (in a number of cases, optimally). Our emphasis is i) the uniform treatment of polyhedra separation problems, ii) the use of hierarchical representations of primitive objects to provide implicit representations of composite or transformed objects, and iii) applications to natural problems in graphics and robotics.

Computing the distance between general convex objects in three-dimensional space

Abstract: A methodology for computing the distance between objects in three-dimensional space is presented. The convex polytope is replaced by a general convex set, avoiding the errors caused by the usual polytope approximations and actually reducing the overall computational time. The basic algorithm is a simple extension of the polytope distance algorithm described by E.G. Gilbert et al. (1988). It utilizes the support mappings of the sets representing the objects. A calculus for evaluating these mappings that allows the extended algorithm to be applied to a rich family of nonpolytopal objects is presented. While the convergence of the algorithm is not finite, it is fast and an effective stopping condition that guarantees the accuracy of the numerical solution is available. Extensive numerical experiments support the claimed efficiency. >

Triangulating a simple polygon in linear time

Abstract: A linear-time deterministic algorithm for triangulating a simple polygon is developed. The algorithm is elementary in that it does not require the use of any complicated data structures; in particular, it does not need dynamic search trees, finger trees, or fancy point location structures. >

Design considerations for a space-variant visual sensor with complex-logarithmic geometry

Abstract: A space-variant sensor design based on the conformal mapping of the half disk, w=log (z+a), with real a>0, which characterizes the anatomical structure of the primate and human visual systems is discussed. There are three relevant parameters: the circumferential index kappa which is defined as the number of pixels around the periphery of the sensor, the visual field radius R (of the half-disk to be mapped), and the map parameter a, which displaces the logarithm's singularity at the origin out of the domain of the mapping. It is shown that the log sensor requires O( kappa /sup 2/log (R/a)) pixels. An analysis is presented which makes it possible to compare directly the space complexity of different sensor designs in the complex logarithmic family. In particular, rough estimates can be obtained of the parameters necessary to duplicate the field width/resolution performance of the human visual system. >

A computational model for face location

Abstract: The authors adopted a model-based approach, where the shape of the object is defined in terms of several mini-templates. The mini-templates are abstract descriptions of simple geometric features like arcs and corners. Relationships between mini-templates are not rigid. Rather, they are represented by springs that allow deformation of a template in terms of its size and orientation. Cost functionals are determined empirically. The authors expect their system to generate candidate regions in a given photograph associated with a rank of its goodness. >

LEDA - A Library of Efficient Data Types and Algorithms

Abstract: LEDA is a library of efficient data types and algorithms. At present, its strength is graph algorithms and related data structures. The computational geometry part is evolving. The main features of the library are •a clear separation of specification and implementation •parameterized data types •its extendibility •its ease of use.

Reconstructing convex sets from support line measurements

Abstract: Algorithms are proposed for reconstructing convex sets given noisy support line measurements. It is observed that a set of measured support lines may not be consistent with any set in the plane. A theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms is developed. The formal statement of the problem and constraints reveals a rich geometry that makes it possible to include prior information about object position and boundary smoothness. The algorithms, which use explicit noise models and prior knowledge, are based on maximum-likelihood and maximum a posteriori estimation principles and are implemented using efficient linear and quadratic programming codes. Experimental results are presented. This research sets the stage for a more general approach to the incorporation of prior information concerning the estimation of object shape. >

A new algorithm for the largest empty rectangle problem

Abstract: A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).

On some link distance problems in a simple polygon

Abstract: A technique is presented for preprocessing a simple polygon to answer link distance queries. The preprocessing requires linear time and the time to triangulate the polygon, and it uses linear storage. As an application of the technique, optimal algorithms for several fundamental link distance problems are derived. >

Querying constraints

Abstract: The design of languages to tackle constraint satisfaction problems has a long history. Only more recently the reverse problem of introducing constraints as primitive constructs in programming languages has been addressed. A main task that the designers and implementers of such languages face is to use and adapt the concepts and algorithms from the extensive studies on constraints done in areas such as Mathematical Programming, Symbolic Computation, Artificial Intelligence, Program Verification and Computational Geometry. In this paper, we illustrate this task in a simple and yet important domain: linear arithmetic constraints. We show how one can design a querying system for sets of linear constraints by using basic concepts from logic programming and symbolic computation, as well as algorithms from linear programming and computational geometry. We conclude by reporting briefly on how notions of negation and canonical representation used in linear constraints can be generalized to account for cases in term algebras, symbolic computation, affine geometry, and elsewhere.

Rapid computation of configuration space obstacles

Abstract: Mathematical properties of configuration space are presented, and algorithms invoking those properties for efficient computation of obstacles in configuration space are described. Simple elements in Cartesian space which can be transformed into configuration space rapidly are identified. Transformations of complex workspace shapes into configuration space are described in terms of multiple transformations of such simpler primitives. Computational considerations and examples are presented for the first three degrees of freedom of an industrial robot. >

Quasi-optimal upper bounds for simplex range searching and new zone theorems

Abstract: This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ℜd so that, given any query simplexq, the points inP ∩q can be counted or reported efficiently. Ifm units of storage are available (n 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.

Nonholonomic motion planning versus controllability via the multibody car system example

Abstract: A multibody car system is a non-nilpotent, non-regular, triangularizable and well-controllable system. One goal of the current paper is to prove this obscure assertion. But its main goal is to explain and enlighten what it means. Motion planning is an already old and classical problem in Robotics. A few years ago a new instance of this problem has appeared in the literature: motion planning for nonholonomic systems. While useful tools in motion planning come from Computer Science and Mathematics (Computational Geometry, Real Algebraic Geometry), nonholonomic motion planning needs some Control Theory and more Mathematics (Differential Geometry). First of all, this paper tries to give a computational reading of the tools from Differential Geometric Control Theory required by planning. Then it shows that the presence of obstacles in the real world of a real robot challenges Mathematics with some difficult questions which are topological in nature, and have been solved only recently, within the framework of Sub-Riemannian Geometry. This presentation is based upon a reading of works recently developed by (1) Murray and Sastry, (2) Lafferiere and Sussmann, and (3) Bellaiche, Jacobs and Laumond.

Parallel geometric algorithms on a mesh-connected computer

Abstract: We show that a number of geometric problems can be solved on a √n × √n mesh-connected computer (MCC) inO(√n) time, which is optimal to within a constant factor, since a nontrivial data movement on an MCC requires Ω(√n) time. The problems studied here include multipoint location, planar point location, trapezoidal decomposition, intersection detection, intersection of two convex polygons, Voronoi diagram, the largest empty circle, the smallest enclosing circle, etc. TheO(√n) algorithms for all of the above problems are based on the classical divide-and-conquer problem-solving strategy.

Grasp synthesis of polygonal objects

Abstract: A simple and efficient algorithm is presented to find the best force closure grasp on a planar polygon with a three-fingered robot hand. In searching for the best grasp, force closure grasps on each feasible combination of edges and/or vertices are constructed using computational geometry. A heuristic function formulated from the consideration of uncertainty in grasping is used to evaluate the quality of each grasp. A test of the algorithm on several different polygons shows that the resulting grasp is realistic and fast enough for real-time use. >

A constrained two-dimensional triangulation and the solution of closest node problems in the presence of barriers

Abstract: A Delaunay triangulation of a set of nodes is a collection of triangles whose vertices are at the nodes and whose union fills the convex hull of the set of nodes. It also has several geometrical properties, making it useful for solving closest point problems. The generalization presented here allows the triangulation to cover nonconvex regions including those with holes. Although a variety of such generalizations are possible, the one presented here is shown to retain important closest point characteristics. Thus it is useful for determining shortest paths within planar regions with polygonal boundaries.

Nonrealizability proofs in computational geometry

Abstract: This paper deals with a class of computational problems in real algebraic geometry. We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries. Hilbert's Nullstellensatz and its real analogue imply that an abstract geometric object is either realizable or it admits a final polynomial. This duality has first been applied by Bokowski in the study of convex polytopes [7] and [11], but in these papers the resulting final polynomials were given without their derivations. It is the objective of the present paper to fill that gap and to describe an algorithm for constructing final polynomials for a large class of nonrealizable chirotopes. We resolve a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence. This result shows that there is no easy combinatorial method for proving nonrealizability and thus justifies our final polynomial approach.

Planar point location revisited

Abstract: Point location is a fundamental primitive in Computational Geometry. In the plane it is stated as follows: Given a subdivision ℛ of the plane and a query point q, determine the region of ℛ containing q. We survey the work that has led to practical algorithms for the static version of the problem, and discuss current research on the corresponding dynamic algorithms.

Subassembly identification and motion generation for assembly: a geometric approach

Abstract: A method for automated generation of an assembly procedure for a given assembly is presented. The procedure is generated from the parts geometry and topology model, and thus explicit specification of mating relations between parts is not required. The approach also prevents wrong mating relations from being presented and makes checking for validity unnecessary. The geometric considerations in the generation of the assembly sequence make it possible to evaluate assembly performance directly and to easily generate detailed assembly plans, such as grasping, transferring, and manipulation, based on the given geometry of the assembly. Algorithms are presented for finding a collision-free path for disassembly motions. The efficiency and feasibility of the algorithms is demonstrated through a case study

Effectively labeling planar projections of polyhedra

Abstract: A well-known method for interpreting planar projections (images) of three-dimensional polyhedra is to label their lines by the Clowes-Huffman scheme. However, the question of whether there is such a labeling has been shown to be NP-complete. A linear-in-time algorithm is given that answers the labelability question under the assumption that some information is known about those edges of the polyhedron both of whose faces are visible. In many cases, this information can be derived from the image itself. Moreover, the algorithm has an effective parallel version, i.e. with polynomially many processors it can be executed in time polynomial in log n. >

Hierarchical triangulation using terrain features

Abstract: Triangulated Irregular Networks (TINs) have been used for surface approximation in many applications. A hierarchical representation is often desirable if the surface is to be rendered at different resolutions. Past work has emphasized techniques where a coarse triangulation is refined by focusing on plane geometry using very little the surface data. For example, a new point is introduced where the deviation of the surface is maximum and the triangle is subdivided into four others. Variants of Delauney triangulations have also been used. We propose a technique where we look more carefully into the features of the surface to be approximated. For example, if there is a ridge, the original triangle is divided by a line along the ridge and one of its vertices is used to subdivided the resulting quadrilateral. In this way the number of very thin triangles (slivers) is significantly reduced. Such triangles produced undesirable effects in animation. In addition the number of levels of the TIN tree is reduced which speeds up searching within the data structure. Tests on data with digital elevation input have confirmed the above theoretical expectations. On eight such sets the average "sliveriness" with the new method was between 1/5 and 1/10 of old triangulations and number of levels was about one third. There was an increase in the number of descendants at each level but the total number of triangles was also lower.Note: Because of space limitations many details and examples have been omitted from this version of the paper. Interested readers should request from the authors a technical report with the same title providing full details of the method, as well as additional examples of implementation than poresented here.