SOCG

In this paper we consider the following problem: Given a set ℒ of n lines in the plane, partition the plane into Ο(r²) triangles so that no triangle intersects more than Ο(n/r) lines of ℒ. We present a deterministic algorithm for this problem with Ο(nr ...

We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary partition is naturally considered as a binary tree where each internal node corresponds to a division and the leaves ...

Paint one side of a rubber disk black and the other side white; stretch the disk any way you wish in three-dimensional space, subject to the condition that from any point in space, if you look down you see either the white side of the disk or nothing at ...

We consider a discrete version of the Whitney-Graustein theorem concerning regular equivalence of closed curves. Two regular polygons P and P', i.e. polygons without overlapping adjacent edges, are called regularly equivalent if there is a continuous ...

The link distance between two points inside a simple polygon P is defined to be the minimum number of edges required to form a polygonal path inside P that connects the points. A link furthest neighbor of a point p Ε P is a point of P whose link ...

We present algorithms that decompose an algebraic curve with rational coefficients in its defining bivariate equation into its irreducible real factors and its non-empty irreducible real components. We show that our algorithms are of polynomial bit ...

The intersection of algebraic curves in three and higher dimensional spaces is considered. An algebrogeometric technique is developed for obtaining an upper bound on the number of intersection points of two irreducible algebraic curves. The asymptotic ...

We present a simple characterization of the lowest degree, implicitly defined, real algebraic surfaces, which smoothly contain any given number of points and algebraic space curves, of arbitrary degree. The characterization is constructive, yielding ...

There is a large and growing body of literature concerning the solution of geometric problems on mesh-connected arrays of processors [5,9,14,17]. Most of these algorithms are optimal (i.e., run in time Ο(n^1/d) on a d-dimensional n-processor array), and ...

We present a parallel algorithm for computing the visible portion of a simple polygonal chain with n vertices from a point in the plane. The algorithm runs in Ο(log n) time using Ο(n/ log n) processors in the CREW-PRAM computational model, and hence is ...

A set A of n points in the plane has to be stored in such a way that for any query triangle t the number of points of A inside t can be computed efficiently. For this problem a solution is presented with Ο(√n log n) query time, Ο (n log n) space and Ο(n^...

Let S ⊂ R³ be an n-set in general position. A plane containing three of the points is called a halving plane if it dissects S into two parts of equal cardinality. It is proved that the number of halving planes is at most Ο(n^2.998).

As a main tool, for ...

Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This ...

We will prove that a finite family B = {B₁, B₂, …, B_n} of connected compact sets in R^d has a hyperplane transversal if and only if for some k there exists a set of points P = {P₁, P₂, …, P_n} (i.e. a k-dimensional labeling of the family) which spans R^k ...

We present an algorithm that solves the following motion-planning problem. Given an L-shaped body L and a 2-dimensional region with n point obstacles, decide whether there is a continuous motion connecting two given positions and orientations of L ...

Given a convex polygon P and an environment consisting of polygonal obstacles, we find the largest similar copy of P that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of ...

We consider motion planning under the compliant motion model, in which a robot directed to walk into a wall may slide along it. We examine several variants of compliant motion planning for a point robot inside a simple polygon with n sides, where the ...

Uncertainty in the execution of robot motion plans must be accounted for in the geometric computations from which plans are obtained, especially in the case where position sensing is inaccurate. We give an Ο(n² log n) algorithm to find a single ...

We present here an algorithm for the curve arrangement problem: determine how a set of planar curves subdivides the plane. This algorithm uses rounded arithmetic and generates an approximate result. It can be applied to a broad class of planar curves, ...

We develop a representation for the topological structure of subdivided manifolds (with and without boundary) of dimension d ≥ 1 which allows straightforward access of the available order information. It is shown that there exists a large amount of ...

This paper deals with the modeling of n-dimensional objects, more precisely with the modeling of subdivisions of n-dimensional topological spaces. We here study the notions of:

• n-dimensional generalized map (or n-G-map), for the modeling of the topology ...

We show, in this paper, how one can probe a class of non convex polyhedra and scenes of disjoint such polyhedra. A polyhedron of that class has convex faces; any two faces are not coplanar and any two edges are not colinear. The basic step of our method ...

We consider a generalization of notions of external visibility of simple polygons, namely weak external visibility, weak external visibility from a line and monotonicity, that we call sector visibility. Informally, sector visibility addresses the ...

We consider the following problem: given a planar set of points S, a measure μ acting on S, and a pair of values μ1 and μ2, does there exist a bipartition S = S₁ U S₂ satisfying μ(S_i) ≤ μ_i for i = 1,2? We present algorithms of complexity Ο(n log n) for ...