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.
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.
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.
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.
43503455
Computing the Extreme Distances Between Two Convex Polygons
Abstract A polygon in the plane is convex if it contains all line segments connecting any two of its points. Let P and Q denote two convex polygons. The computational complexity of finding the minimum and maximum distance possible between two points p in P and q in Q is studied. An algorithm is described that determines the minimum distance (together with points p and q that realize it) in O (log m + log n ) time, where m and n denote the number of vertices of P and Q , respectively. This is optimal in the worst case. For computing the maximum distance, a lower bound Ω(m + n) is proved. This bound is also shown to be best possible by establishing an upper bound of O ( m + n ).
22962151
On the Number of Line Separations of a Finite Set in the Plane
Abstract Let S denote a set of n points in the Euclidean plane. A subset S′ of S is termed a k-set of S if it contains k points and there exists a straight line which has no point of S on it and separates S′ from S−S′. We let fk(n) denote the maximum number of k-sets which can be realized by a set of n points. This paper studies the asymptotic behaviour of fk(n) as this function has applications to a number of problems in computational geometry. A lower and an upper bound on fk(n) is established. Both are nontrivial and improve bounds known before. In particular, f k (n) = f n−k (n) = Ω(n log k) is shown by exhibiting special point-sets which realize that many k-sets. In addition, f k (n) = f n−k (n) = O(nk 1 2 ) is proved by the study of a combinatorial problem which is of interest in its own right.
9524011
On the Definition and the Construction of Pockets in Macromolecules
The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known three-dimensional conformations.
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.