10864618
Stability of persistence diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.
17520762
Extraction and simplification of iso-surfaces in tandem
The tandem algorithm combines the marching cube algorithm for surface extraction and the edge contraction algorithm for surface simplification in lock-step to avoid the costly intermediate step of storing the entire extracted surface triangulation. Beyond this basic strategy, we introduce refinements to prevent artifacts in the resulting triangulation, first, by carefully monitoring the amount of simplification during the process and, second, by driving the simplification toward a compromise between shape approximation and mesh quality. We have implemented the algorithm and used extensive computational experiments to document the effects of various design options and to further fine-tune the algorithm.
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.
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.
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.
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.
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.
16373684
Counting triangle crossings and halving planes
Every collection of t ≥ 2n<supscrpt>2</supscrpt> triangles witha total of n vertices in <inline-equation><f><blkbd>R</blkbd><sup>3</sup></f></inline-equation> has <inline-equation><f><g>W</g><fen lp="par"><fr><nu>t<sup>4</sup></nu><de>n<sup>6</sup></de></fr><rp post="par"></fen></f></inline-equation> crossing pairs. This implies that one of their edgesmeets<inline-equation><f><g>W</g><fen lp="par"><fr><nu>t<sup>3</sup></nu><de>n<sup>6</sup></de></fr><rp post="par"></fen></f><?Pub Caret></inline-equation> of the triangles. From this it followsthat n points in <inline-equation><f><blkbd>R</blkbd><sup>3</sup></f></inline-equation> have only O(n<supscrpt>8/3</supscrpt>) halvingplanes.