Chebychev Approximations of Finite Point Sets with Application to Planar Kinematic Synthesis

1979 ◽  
Vol 101 (1) ◽  
pp. 32-40 ◽  
Author(s):  
Y. L. Sarkisyan ◽  
K. C. Gupta ◽  
B. Roth
Keyword(s):  
Point Sets ◽  
Finite Point ◽  
Kinematic Synthesis ◽  
Radial Deviation ◽  
Point Set ◽  
Planar Linkages ◽  
Straight Lines ◽  
Set Of Points

In the first part of this paper we consider the problem of determining circles which best approximate a given set of points. The approximation is one which minimizes the maximum radial deviation of the points from the approximating circle. Then a similar procedure is developed for determining straight lines which best approximate a given point set. The final parts of the paper illustrate the application of these results to synthesizing planar linkages.

EMBEDDING POINT SETS INTO PLANE GRAPHS OF SMALL DILATION

2007 ◽  
Vol 17 (03) ◽  
pp. 201-230 ◽  
Author(s):  
ANNETTE EBBERS-BAUMANN ◽  
ANSGAR GRÜNE ◽  
ROLF KLEIN ◽  
MAREK KARPINSKI ◽  
CHRISTIAN KNAUER ◽  
...  
Keyword(s):  
Plane Graph ◽  
Upper And Lower Bounds ◽  
Convex Curve ◽  
Embedding Problem ◽  
Finite Point ◽  
Plane Graphs ◽  
Parallel Lines ◽  
Vertex Set ◽  
Point Set ◽  
Set Of Points

Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound > 1. In this paper we provide the first upper and lower bounds for the embedding problem. 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. 2. Each embedding of a closed convex curve has dilation ≥ 1.00157. 3. Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation [Formula: see text].

scholarly journals Induced Ramsey-Type Results and Binary Predicates for Point Sets

10.37236/7039 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Martin Balko ◽  
Jan Kynčl ◽  
Stefan Langerman ◽  
Alexander Pilz
Keyword(s):  
Order Type ◽  
Point Sets ◽  
Finite Point ◽  
Local Consistency ◽  
New Family ◽  
Consistency Property ◽  
Ramsey Type ◽  
Point Set ◽  
Positive Integers ◽  
Ordered Pairs

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey.As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.

scholarly journals A Bound on a Convexity Measure for Point Sets

2019 ◽  
Vol 29 (04) ◽  
pp. 301-306
Author(s):  
Danny Rorabaugh
Keyword(s):  
Upper Bound ◽  
Interior Angle ◽  
Point Sets ◽  
Planar Point ◽  
Angle Measure ◽  
Convex Position ◽  
Point Set ◽  
Set Of Points ◽  
Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

Two Algorithms for the Sum of Diameters Problem and a Related Problem

1997 ◽  
Vol 07 (05) ◽  
pp. 493-508
Author(s):  
Muhammad H. Alsuwaiyel
Keyword(s):  
Real Number ◽  
Related Problem ◽  
Point Sets ◽  
Point Set ◽  
Almost All ◽  
Set Of Points

Given a set S of points in the plane, we consider the problem of partitioning S into two subsets such that the sum of their diameters is minimized. We present two algorithms with time complexities O(n log 2 n / log log n) and O(n log n / (1 - ∊)), where ∊, 0 < ∊ < 1, is a real number that is dependent on the density of the point set. In almost all practical instances, the second algorithm runs in optimal O(n log n) time, improving all previous results in the case of nonsparse point sets. These bounds follow immediately from two corresponding algorithms with the same time complexities for the following problem: given a set of points S = {p1, p2, …,pn} in the plane sorted in increasing distance from p1, compute the sequence of diameters d1, d2, …, dn, where di= Diam {p1, …, pi} is the diameter of the first i points, 1 ≤ i ≤ n.

scholarly journals Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

2018 ◽  
Vol 13 (1) ◽  
pp. 65-86 ◽  
Author(s):  
Mario Neumüller ◽  
Friedrich Pillichshammer
Keyword(s):  
Quantitative Measure ◽  
Explicit Construction ◽  
Unit Cube ◽  
Point Sets ◽  
Finite Point ◽  
Quasi Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Dimensional Unit ◽  
Point Set ◽  
Star Discrepancy

Abstract The star discrepancy $D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and $D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$ . However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed−1). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and $D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$ . Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (nα)-sequence and showed a metrical discrepancy bound of the form $C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C> 0 independent of N and d. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

scholarly journals MAINTAINING VISIBILITY INFORMATION OF PLANAR POINT SETS WITH A MOVING VIEWPOINT

2007 ◽  
Vol 17 (04) ◽  
pp. 297-304 ◽  
Author(s):  
OLIVIER DEVILLERS ◽  
VIDA DUJMOVIĆ ◽  
HAZEL EVERETT ◽  
SAMUEL HORNUS ◽  
SUE WHITESIDES ◽  
...  
Keyword(s):  
Linear Space ◽  
Point Sets ◽  
Worst Case ◽  
Planar Point ◽  
Set Of Points

Given a set of n points in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint q and efficiently maintaining this ordering information as q moves. In linear space, and after O(n log n) preprocessing time, our solution maintains the view at a cost of O( log n) amortized time (resp.O( log 2 n) worst case time) for each change. Our algorithm can also be used to maintain the set of points sorted according to their distance to q .

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