SOCG

We will consider an arrangement H of n hyperplanes in E^d (where the dimension d is fixed). An ε-cutting for H will be a collection of (possibly unbounded) d-dimensional simplices with disjoint interiors, which cover all E^d and such that the interior of ...

Given any natural number d, 0 < ε < 1, let ƒ_d(ε) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an ε-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒ_d(ε) > ...

It is known that in general range spaces of VC-dimension d > 1 require ε-nets to be of size at least Ω(d/ε log 1/ε). We investigate the question whether this general lower bound is valid for the special range spaces that typically arise in computational ...

We give a new Ο(n log log n)-time deterministic linear-time algorithm for triangulating simple n-vertex polygons, which avoids the use of complicated data-structures. In addition, for polygons whose vertices have integer coordinates of polynomially ...

We show that a triangulation of a set of n points in the plane that minimizes the maximum angle can be computed in time O(n² log n) and space O(n). In the same amount of time and space we can also handle the constrained case where edges are prescribed. ...

A number of problems in computational geometry involving simple polygons can be solved in linear time once the polygon has been triangulated. Since the worst-case time bound for triangulating a general simple polygon is currently super-linear, these ...

Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that ...

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest path problems for simple polygons. Our algorithms all run in Ο(log n) time and are based on the use of a new data structure for implicitly representing ...

We show that the maximum combinatorial complexity of the space of hyperplane transversals to a family of n separated and strictly convex sets in R^d is Θ(n^⌊d/2⌋), which generalizes results of Edelsbrunner and Sharir in the plane. As a key step in the ...

We investigate the computational problems associated with combinatorial surfaces. Specifically, we present an algorithm (based on the Brahana-Dehn-Heegaard approach) for transforming the polygonal schema of a closed triangulated surface into its ...

We prove that for any set S of n points in the plane and n^3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n^3-3α/(512 log⁵ n) of the triangles. This implies that any set of n points in three-...

We show that for any set Π of n points in three-dimensional space there is a set Q of 𝒪(n^1/2 log³ n) points so that the Delaunay triangulation of Π ∪ Q has at most 𝒪(n^3/2 log³ n) edges — even though the Delaunay triangulation of Π may have Ω(n²) ...

If a closed curve in space is a trivial knot (intuitively, one can untie it without cutting) then it is the boundary of some disk with no self-intersections. In this paper we investigate the minimum number of faces of a polyhedral spanning disk of a ...

The set of solutions to a collection of polynomial equations is referred to as an algebraic set. An algebraic set that cannot be represented as the union of two other distinct algebraic sets, neither containing the other, is said to be irreducible. An ...

We consider the problem of computing a piecewise linear approximation to the intersection of a pair of tensor product B-spline surfaces in 3-space. The problem is rather central in solid modeling. We present a fast and robust divide-and-conquer ...

Let V be a set of objects in space for which we want to determine the portions visible from a particular point of view υ. Assume V is subdivided in subsets V₁,…, V_z and the visibility maps Μ₁, … Μ_z of these subsets from point ugr; are known. We show ...

In this paper we consider the following problems: given a set T of triangles in 3-space, with |T| = n,

• answer the query “given a line l, does l stab the set of triangles?” (query problem).

• find whether a stabbing line exists for the set of triangles (...

A K-d tree represents a set of N points in K-dimensional space. Operations on a semidynamic tree may delete and undelete points, but may not insert new points. This paper shows that several operations that require Ο(log N) expected time in general K-d ...

We give a polynomial approximation sceme for subdividing a simple polygon into approximately equal parts by curves of the smallest possible total length. For convex polygons we show that an exact fast algorithm is possible. Several generalizations are ...

We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of n points in @@@@^d in time 𝒪(Τ_d(N, N) log^d N), where Τ_d(n, m) is the time required to compute a bichromatic closest pair among n red and m blue points in @@@@^d. If ...

We present two randomized algorithms. One solves linear programs involving m constraints in d variables in expected time Ο(m). The other constructs convex hulls of n points in R^d, d > 3, in expected time Ο(n^⌈d/2⌉). In both bounds d is considered to be a ...

For compact Euclidean bodies P, Q, we define λ(P, Q) to be smallest ratio r/s where r > 0, s > 0 satisfy sQ′ ⊆ P ⊆ rQ″. Here sQ denotes a scaling of Q by factor s, and Q′, Q″ are some translates of Q. This function λ gives us a new distance function ...

This paper considers the maximin placement of a convex polygon P inside a polygon Q, and introduce several new static and dynamic Voronoi diagrams to solve the problem. It is shown that P can be placed inside Q, using translation and rotation, so that ...

One useful generalization of the convex hull of a set S of points is the ε-strongly convex δ-hull. It is defined to be a convex polygon Rgr; with vertices taken from S such that no point in S lies farther than δ outside Rgr; and such that even if the ...

A standard technique in solid modeling is to represent planes (or lines) by explicit equations and to represent vertices and edges implicitly by means of combinatorial information. Numerical problems that arise from using floating-point arithmetic to ...

This paper presents a connection between the problem of drawing a graph with minimum number of edge crossings, and the theory of arrangements of pseudolines. In particular, we show that any given arrangement can be forced to occur in every minimum-...

We study enumeration and visibility problems in the d-dimensional integer lattice L^d_n of d-tuples of integers ≤ n. In the first part of the paper we give several useful enumeration principles and use them to study the asymptotic behavior of the number ...