34884676
Cutting dense point sets in half
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 halfspaces. 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 δn(1/d), δ 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 Ω (n log n), 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.
12111408
Cutting dense point sets in half
A <italic>halving hyperplane</italic> of a set <italic>S</italic> of <italic>n</italic> points in <bold>R</bold><supscrpt><italic>d</italic></supscrpt> contains <italic>d</italic> affinely independent points of <italic>S</italic> 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 δ<italic>n</italic><supscrpt>1/<italic>d</italic></supscrpt>, δ some constant. Such a set <italic>S</italic> is called <italic>dense</italic>.
In <italic>d</italic>=2 dimensions, the number of halving lines for a dense set can be as much as &OHgr;(<italic>n</italic>log<italic>n</italic>), and it cannot exceed <italic>O</italic>(<italic>n</italic><supscrpt>5/4</supscrpt>/log*<italic>n</italic>). The upper bound improves over the current best bound of <italic>O</italic>(<italic>n</italic><supscrpt>3/2</supscrpt>/log*<italic>n</italic>) which holds more generally without any density assumption. In <italic>d</italic>=3 dimensions we show that <italic>O</italic>(<italic>n</italic><supscrpt>7/3</supscrpt>) 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 <italic>d</italic>≥4 dimensions, where it leads to <italic>O</italic>(<italic>n</italic><supscrpt><italic>d</italic>−2/<italic>d</italic></supscrpt>) as an upper bound for the number of halving hyperplanes.
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.
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.
14707519
Inequalities for the curvature of curves and surfaces
In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R3 in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.
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.
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.