SOCG

Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments ...

We give a simple algorithmic technique for building geometric structures. The technique is randomized and incremental. As an application, we give an algorithm of this kind for computing the intersection of a set of halfspaces in three dimensions. (This ...

We present an algorithm that triangulates a simple polygon on n vertices in Ο(n log^* n) expected time. The algorithm uses random sampling on the input, and its running time does not depend on any assumptions about a probability distribution from which ...

The range searching problems which allow partition trees where every query enters only a sublinear number of nodes are characterized as those with finite Vapnik - Chervonenk is dimension.

The concrete combinatorial bounds obtained imply—among others — ...

We define a new formalism called the quasi-valid range aggregation. This formalism leads to a new and quite simple method for reducing non-range query-like problems to range queries and often to orthogonal range queries, with immediate applications to ...

We show that the total number of edges of m faces of an arrangement of n lines in the plane is Ο(m^2/3-δ n^2/3+2δ + n), for any δ > 0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of these m faces and ...

An arrangement of n lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists of Ο(n²) regions, called faces. In this paper we study the problem of calculating and storing arrangements ...

Let Γ be a collection of n (possibly intersecting) “red” Jordan arcs of some simple shape in the plane and let Γ' be a similar collection of m “blue” arcs. We present several efficient algorithms for detecting an intersection between an arc of Γ and an ...

In this lecture we present a survey on three algebraic algorithmic methods for problems in algebraic geometry: Characteristic Sets (Ritt, Wu), Gröbner Bases (Buchberger), and Cylindrical Algebraic Decomposition (Collins).

There are at least three reasons ...

We describe the design of LINETOOL, a geometric editor. Researchers in the areas of computational geometry, robotics and algebraic computation need a graphical editor for composing geometric objects which does more than simply turn pixels on and off on ...

Geometric computations, like all numerical procedures, are extremely prone to roundoff error. However, virtually none of the numerical analysis literature directly applies to geometric calculations. Even for line intersection, the most basic geometric ...

This paper describes a general purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task to provide a consistent treatment ...

In a previous paper, we introduced a generic solution to the problem of data degeneracy in geometric algorithms. The scheme is simple to use: algorithms qualifying under our requirements just have to use a prescribed blackbox for polynomial evaluation ...

This paper is concerned with a simple, rather intuitive preprocessing step that is likely to improve the average-case performance of any convex hull algorithm. For n points randomly distributed in the unit square, we show that a simple linear pass ...

Let S be a set of n non-intersecting line segments in the plane. The visibility graph G_S of S is the graph that has the endpoints of the segments in S as nodes and in which two nodes are adjacent whenever they can “see” each other (i.e., the open line ...

The problem of determining the Euclidean shortest path between two points in the presence of m simple polygonal obstacles is studied. An O( m² logn + nlogn ) algorithm is developed, where n is the total number of points in the obstacles. A simple O(E+T) ...

A simple but important special case of the hidden surface removal problem is one in which the scene consists of n rectangles with sides parallel to the x and y-axes, with viewpoint at z = ∞ (that is, an orthographic projection). This special case has ...

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running ...

In this paper we give parallel algorithms for a number of problems defined on polygons and point sets. All of our algorithms have optimal T(n) ^* P(n) products, where T(n) is the time complexity and P(n) is the number of processors used, and are for the ...

We consider the problem of covering simple orthogonal polygons with star polygons. 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.

In general, orthogonal ...

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of ...

An algorithm is presented for finding a maximum-weight spanning tree of a set of n points in the Euclidean plane, where the weight of an edge (pi, pj) equals the Euclidean distance between the points pi and pj. The algorithm runs in time Ο (n logn) and ...

We consider clustering problems under two different optimization criteria. One is to minimize the maximum intracluster distance (diameter), and the other is to maximize the minimum intercluster distance. In particular, we present an algorithm which ...

Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union of n regions bounded by Jordan curves in the plane. We show that, under some fairly weak ...

We consider the terrain navigation problem in a two-dimensional polygonal subdivision consisting of obstacles, “free” regions (in which one can travel at no cost), and regions in which cost is proportional to distance traveled. This problem is a special ...

Motion planning algorithms have generally dealt with motion in a static environment, or more recently, with motion in an environment that changes in a known manner. We consider the problem of finding collision-free motions in a changeable environment. ...

We show that, under reasonable assumptions, any collision-avoiding motion planning problem for a moving system with two degrees of freedom can be solved in time Ο(λ_s(n) log²n), where n is the number of collision constraints imposed on the system, s is a ...