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.
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.
2265636
Polygonal Intersection Searching
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.
17674541
Some methods of computational geometry applied to computer graphics
Abstract Windowing a two-dimensional picture means to determine those line segments of the picture that are visible through an axis-parallel window. A study of some algorithmic problems involved in windowing a picture is offered. Some methods from computational geometry are exploited to store the picture in a computer such that (1) those line segments inside or partially inside of a window can be determined efficiently, and (2) the set of those line segments can be maintained efficiently while the window is moved parallel to a coordinate axis and/or it is enlarged or reduced.
33731652
Rectangular Point Location in d Dimensions with Applications
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.
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.
1993090
Morse-smale complexes for piecewise linear 3-manifolds
We define the Morse-Smale complex of a Morse function over a 3-manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3-dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.
30416185
On the Zone Theorem for Hyperplane Arrangements
The zone theorem for an arrangement of n hyperplanes in d-dimensional real space says that the total number of faces bounding the cells intersected by another hyperplane is $O(n^{d - 1} )$. This result is the basis of a time-optimal incremental algorithm that constructs a hyperplane arrangement and has a host of other algorithmic and combinatorial applications. Unfortunately, the original proof of the zone theorem, for $d \geqslant 3$, turned out to contain a serious and irreparable error. This paper presents a new proof of the theorem. The proof is based on an inductive argument, which also applies in the case of pseudohyperplane arrangements. The fallacies of the old proof along with some ways of partially saving that approach are briefly discussed.
444430
Loops in reeb graphs of 2-manifolds
Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(nlogn), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.