7249846
Space Searching for Intersecting Objects
Abstract 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, planes, and polyhedra in three dimensions.
30538700
Computing a Ham-Sandwich Cut in Two Dimensions
Let B be a set of n"b black points and W a set of n"w, white points in the Euclidean plane. A line h is said to bisect B (or W) if, at most, half of the points of B (or W) lie on any one side of h. A line that bisects both B and W is called a ham-sandwich cut of B and W. We give an algorithm that computes a ham-sandwich cut of B and W in 0((n"h+n"w) log (min {n"b, n"w}+ 1)) time. The algorithm is considerably simpler than the previous most efficient one which takes 0((n"b + n"w) log (n"b + n"w)) time.
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.
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.
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.
2265636
Polygonal Intersection Searching
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.
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.
44420378
Triangulating Topological Spaces
Given a subspace ${\Bbb X} \subseteq {\Bbb R}^d$ and a finite set S⊆ℝd, we introduce the Delaunay complex, ${\cal D}_{\Bbb X}$, restricted by ${\Bbb X}$. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets ${\Bbb X}$ in a non-empty set. By the nerve theorem, $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ be homeomorphic.
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.
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.