1600979
Three-dimensional alpha shapes
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 α-shapes of a finite point set in R<supscrpt>3</supscrpt>. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter α ε R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size <italic>n</italic> in time <italic>0(n<supscrpt>2</supscrpt>)</italic>, worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing  are mentioned.
9599628
The union of balls and its dual shape
Efficient algorithms are described for computing topological,combinatorial, and metric properties of the union of finitely many ballsin <inline-equation><f><blkbd>R</blkbd><sup>d</sup></f></inline-equation>. These algorithms are based on a simplicial complexdual to a certain decomposition of the union of balls, and on shortinclusion-exclusion formulas derived from this complex. The algorithmsare most relevant in <inline-equation><f><blkbd>R</blkbd><sup>3</sup></f></inline-equation> <?Pub Caret>where unions of finitely many balls arecommonly used as models of molecules.
6983029
On the shape of a set of points in the plane
A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.
53237055
Three-dimensional alpha shapes
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 α-shapes of a finite point set in R 3 . Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter α 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(n 2 ) , worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.
46726120
The union of balls and its dual shape
Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in ℝd. These algorithms are based on a simplicial complex dual to a decomposition of the union of balls using Voronoi cells, and on short inclusion-exclusion formulas derived from this complex. The algorithms are most relevant in ℝ3 where unions of finitely many balls are commonly used as models of molecules.
12582971
Stabbing line segments
An algorithm for the geometric problem of determining a line (called a stabbing line) which intersects each ofn given line segments in the plane is presented. As a matter of fact, the algorithm computes a description of all stabbing lines. A purely geometric fact is proved which infers that this description requiresO(n) space to be specified. Our algorithm computes it inO(n logn) time which is optimal in the worst case.Using the description of the stabbing lines, we are able to decide inO(logn) time whether or not a specified line is a stabbing line. Finally, the problem of maintaining the description of all stabbing lines while inserting and deleting line segments is addressed.
17780302
Triangulating topological spaces
Given a subspace<inline-equation><f><blkbd>X⊆R</blkbd><sup>d</sup></f></inline-equation> and a finite set <inline-equation><f>S⊆<blkbd>R</blkbd><sup>d</sup></f></inline-equation>, we introduce the Delaunay simplicial complex, <inline-equation><f><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation>, restricted by <inline-equation><f><blkbd>X</blkbd></f></inline-equation>. Its simplices are spanned by subsets<inline-equation><f>T⊆S</f></inline-equation> for which the common intersection of Voronoi cells meets <inline-equation><f><blkbd>X</blkbd></f></inline-equation> in a non-empty set. By the nerve theorem,<inline-equation><f><lim align="c"><op>⋃</op></lim><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation> and <inline-equation><f><blkbd>X</blkbd></f></inline-equation> are homotopy equivalent if all such sets are contractible. This paper shows that <inline-equation><f><lim align="c"><op>⋃</op></lim><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation> and <inline-equation><f><blkbd>X</blkbd></f></inline-equation> <?Pub Caret1>are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.
6301471
Adaptive Simplicial Grids from Cross-Sections of Monotone Complexes
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.
31366541
An optimal algorithm for constructing the weighted voronoi diagram in the plane
Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by de(x,p)/w(p) where de denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S. 
 
An algorithm which constructs the weighted Voronoi diagram for S in O(n2) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n2) faces, edges and vertices.
41540520
Space Searching for Intersecting Objects
Determining or counting geometric objects that intersect another geometric query object is at the core of algorithmic problems in a number of applied areas of computer science. This article presents a family of space-efficient data structures that realize sublinear query time for points, line segments, lines and polygons in the plane, and points, line segments, plaraes, and polyhedra in three dimensions.