Showing papers by "Herbert Edelsbrunner published in 1992"

Three-dimensional alpha shapes

Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of a-shapes of a finite point set in R3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter a e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n2), worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.

An optimal algorithm for intersecting line segments in the plane

Abstract: The main contribution of this work is an O(n log n + k)-time algorithm for computing all k intersections among n line segments in the plane. This time complexity is easily shown to be optimal. Within the same asymptotic cost, our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.

Incremental topological flipping works for regular triangulations

Abstract: A set ofn weighted points in general position in ℝ d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

Diameter, width, closest line pair, and parametric searching

Abstract: We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them. We obtain, for any fixed e > 0, an O(n1+e) algorithm for computing the diameter of a point set in 3-space, an O(n8/5+e) algorithm for computing the closest pair in a set of n lines in space. All these algorithms are deterministic. We also look at the problem of computing the k-th smallest slope formed by the lines joining n points in the plane. In 1989 Cole, Salowe, Steiger, and Szemere´di gave an optimal but very complicated O(n log n) solution based on Megiddo's technique. We follow a different route and give a very simple O(n log2n) solution which bypasses parametric searching altogether.

An O ( n 2 log n ) time algorithm for the minmax angle triangulation

Abstract: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum angle can be computed in time $O(n^2 \log n)$ and space $O(n)$. The algorithm is fairly easy to implement and is based on the edge-insertion scheme that iteratively improves an arbitrary initial triangulation. It can be extended to the case where edges are prescribed, and, within the same time- and space-bounds, it can lexicographically minimize the sorted angle vector if the point set is in general position. Experimental results on the efficiency of the algorithm and the quality of the triangulations obtained are included.

Counting and cutting cycles of lines and rods in space

Abstract: A number of rendering algorithms in computer graphics sort three-dimensional objects by depth and assume that there is no cycle that makes the sorting impossible. One way to resolve the problem caused by cycles is to cut the objects into smaller pieces. The problem of estimating how many such cuts are always sufficient is addressed. A few related algorithmic and combinatorial geometry problems are considered. >

An upper bound for conforming Delaunay triangulations

Abstract: A plane geometric graph C in R2conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m2n) points that conforms to G. The algorithm that constructs the points is also described.

The number of edges of many faces in a line segment arrangement

Abstract: We show that the maximum number of edges boundingm faces in an arrangement ofn line segments in the plane isO(m2/3n2/3+nα(n)+nlogm). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m2/3n2/3+nα(n)). In addition, we show that the number of edges bounding anym faces in an arrangement ofn line segments with a total oft intersecting pairs isO(m2/3t1/3+nα(t/n)+nmin{logm,logt/n}), almost matching the lower bound of Ω(m2/3t1/3+nα(t/n)) demonstrated in this paper.

Edge Insertion for Optional Triangulations

Abstract: The edge-insertion paradigm improves a triangulation of a finite point set in ℜ2 iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.