Showing papers by "Herbert Edelsbrunner published in 1994"

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.

Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms

Abstract: 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 for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those that do not use it. We believe that this technique will become a standard tool in writing geometric software.

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 paper introduces the formal notion of the family of $\alpha$-shapes of a finite point set in $\Real^3$. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter $\alpha \in \Real$ 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 $O(n^2)$, worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.

Ray shooting in polygons using geodesic triangulations

Abstract: LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

Triangulating topological spaces

Abstract: Given a subspace 𝒳 ⊆ Rd and a finite set S⊆Rd, we introduce the Delaunay simplicial complex, D𝒳, restricted by 𝒳. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets 𝒳 in a non-empty set. By the nerve theorem,⋃D𝒳 and 𝒳 are homotopy equivalent if all such sets are contractible. This paper shows that ⋃D𝒳 and 𝒳 are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.

Algorithms for bichromatic line-segment problems and polyhedral terrains

Abstract: We consider a variety of problems on the interaction between two sets of line segments in two and three dimensions. These problems range from counting the number of intersecting pairs between m blue segments andn red segments in the plane (assuming that two line segments are disjoint if they have the same color) to finding the smallest vertical distance between two nonintersecting polyhedral terrains in three-dimensional space. We solve these problems efficiently by using a variant of the segment tree. For the three-dimensional problems we also apply a variety of recent combinatorial and algorithmic techniques involving arrangements of lines in three-dimensional space, as developed in a companion paper.

Counting triangle crossings and halving planes

Abstract: Every collection oft?2n2 triangles with a total ofn vertices in ?3 has Ω(t4/n6) crossing pairs. This implies that one of their edges meets Ω(t3/n6) of the triangles. From this it follows thatn points in ?3 have onlyO(n8/3) halving planes.

Cutting dense point sets in half

Abstract: A halving hyperplane of a set S of n points in Rd contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most dn1/d, d some constant. Such a set S is called dense.In d=2 dimensions, the number of halving lines for a dense set can be as much as O(nlogn), and it cannot exceed O(n5/4/log*n). The upper bound improves over the current best bound of O(n3/2/log*n) which holds more generally without any density assumption. In d=3 dimensions we show that O(n7/3) is an upper bound on the number of halving planes for a dense set. The proof is based on a metric argument that can be extended to d≥4 dimensions, where it leads to O(nd−2/d) as an upper bound for the number of halving hyperplanes.

Selecting Heavily Covered Points

Abstract: A collection of geometric selection lemmas is proved, such as the following: For any set $P$ of $n$ points in three-dimensional space and any set ${\cal S}$ of $m$ spheres, where each sphere passes through a distinct point pair in $P$, there exists a point $x$, not necessarily in $P$, that is enclosed by $\Omega (m^2/(n^2 \log^6 {n^2 \over m}))$ of the spheres in ${\cal S}$. Similar results apply in arbitrary fixed dimensions, and for geometric bodies other than spheres. The results have applications in reducing the size of geometric structures, such as three-dimensional Delaunay triangulations and Gabriel graphs, by adding extra points to their defining sets.

Adaptive simplicial grids from cross-sections of monotone complexes

Abstract: We study the maintenance of a simplicial grid or complex under changing density requirements The proposed method works in any fixed dimension and generates grids by projecting cross-sections of a monotone simplicial complex that lives in one dimension higher than the grid The density of the grid is adapted by locally moving the cross-section up or down along the extra dimension