Finite Planar Set Research Articles

Slime mould computes planar shapes

Computing a polygon defining a set of planar points is a classical problem of modern computational geometry. In laboratory experiments, we demonstrate that a concave hull, a connected α-shape without holes, of a finite planar set is approximated by slime mould Physarum polycephalum. We represent planar points with sources of long-distance attractants and short-distance repellents and inoculate a piece of plasmodium outside the dataset. The plasmodium moves towards the data and envelops it by pronounced protoplasmic tubes.

Large convexly independent subsets of Minkowski sums

Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $\mathbf{R}^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\geq \Omega(n\sqrt{\log n})$, which answers a question of Eisenbrand, Pach, Rothvoß, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\lfloor\frac{1}{3}n^2\rfloor\leq E_3(n)\leq \frac{3}{8}n^2+O(n^{3/2})$. Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs in a set of $n$ points in some $d$-dimensional strictly convex normed space. We show that $W_2(n)=\Theta(E_2(n))$ and for $d\geq 3$ that $W_d(n)\sim\frac12\left(1-\frac{1}{a(d)}\right)n^2$, where $a(d)\in\mathbf{N}$ is related to strictly antipodal families. In fact we show that the same asymptotics hold without the requirement that the unit distance pairs form pairwise nonparallel segments, and also if diameter pairs are considered instead of unit distance pairs.

Method of orienting curves for determining the convex hull of a finite set of points in the plane

In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung in Großen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Graham's convex hull algorithm are presented.

On 5-isosceles sets in the plane, with linear restriction

A finite planar set is k-isosceles for k≥3 if every k-point subset of the set contains a point equidistant from two others. We show that an 8-set on a line is 5-isosceles if and only if its adjacent interpoint distances are equal to each other, and no 5-isosceles 9-set has 9 points on a line. We also show that the maximum 5-isosceles set with 8 points on a line contains at most 10 points.

Computational and approximational complexity of combinatorial problems related to the committee polyhedral separability of finite sets

In [1, 2], results on the computational and approximational complexity of a minimum affine separating committee (MASC) problem were obtained for finite sets A, B ⊂ ℚn . In particular, it was shown that this problem is NP-hard and does not belong to the class Apx (under the assumption that P ≠ NP). Nevertheless, questions concerning the bounds for its effective approximability threshold and for the computational complexity of a number of practically important particular cases of the problem obtained by imposing additional constraints, for example, by fixing the dimension of the space, remained open. In this paper, a lower bound is presented for the polynomial approximability threshold of the problem in the general case, and the intractability of the problem in spaces of fixed dimension greater than unity is proved. In particular, it is shown that the problem of committee separability remains hard even when it is formulated on the plane (i.e., in the simplest non-trivial case). This result follows from the fact that the well-known PC problem on covering a finite planar set by straight lines, whose hardness was proved in [3], is polynomially reducible to the problem under consideration. The method of reduction represents a modification of the method that was described in [4] and was used there for proving the hardness of problems on piecewise linear separability of finite sets on the plane.

More About 4-Isosceles Planar Sets

A finite planar set is k-isosceles for k ≥ 3 if every k-point subset of the set contains a point equidistant from the other two. In [1] Fishburn obtains several important results about isosceles planar sets and poses a series of conjectures and open questions. We disprove Conjecture 1 in [1] and provide another 34 nonsimilar 4-isosceles 8-point planar sets which answer one of the open questions in [1] negatively.

About 4-Isosceles Planar Sets

A finite planar set is k-isosceles for k \geq 3, if every k -point subset of the set contains a point equidistant from the other two. This paper gives affirmative answers to a few open problems about 4-isosceles sets.

A connectionist model for convex-hull of a planar set

A neural network model is proposed for computation of the convex-hull of a finite planar set. The model is self-organizing in that it adapts itself to the hull-vertices of the convex-hull in an orderly fashion without any supervision. The proposed network consists of three layers of processors. The bottom layer computes the activation functions, the outputs of which are passed onto the middle layer. The middle layer is used for winner selection. These information are passed onto the topmost layer as well as fed back to the bottom layer. The network in the topmost layer self-organizes by labeling the hull-processors in an orderly fashion so that the final convex-hull is obtained from the topmost layer. Time complexities of the proposed model are analyzed and are compared with existing models of similar nature.

Isosceles Planar Subsets

A finite planar set is k -isosceles for k ≥ 3 if every k -point subset of the set contains a point equidistant from two others. There are three nonsimilar 3-isosceles sets with five points and one with six points. Eleven 4-isosceles sets with eight points are noted, and it is conjectured that no 4-isosceles set has nine points. Exactly one 4-isosceles 8-set has four points on a line, and every 4-isosceles set that includes the vertices of a square has fewer than eight points. 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p391.pdf yes no no yes

Distinct distances in finite planar sets

For each n ⩾ 3 let Fn denote the set of all integer vectors f = (f1, f2, …, fn) with 1 ⩽ f1 ⩽ f2 ⩽ ··· ⩽ fn ⩽ n − 1 for which there exists a set x1, x2, …, xn of n points in the plane such that each xi has exactly fi different distances to the other n − 1 points. Thus, F3 = n0 ⊮(1, 1, 1), (1, 2, 2), (2, 2, 2) . We determine all n-point configurations for n ⩽ 7 that minimize ∑fi over Fn and show that min Larr; bEfi: f ∈ F8gsim; = 24. We note that every small n except n ∈ 8, 9 has a subset of the triangular lattice amont its sum-minimizing configurations, and conjecture that subsets of the lattice minimize ∑fi for all large n.

The two largest distances in finite planar sets

We determine all homogenous linear inequalities satisfied by the numbers of occurrences of the two largest distances among n points in the plane.

Multiplicities of interpoint distances in finite planar sets

What is the maximum number of unit distances between the vertices of a convex n-gon in the plane? We review known partial results for this and other open questions on multiple occurrences of the same interpoint distance in finite planar subsets. Some new results are proved for small n. Challenging conjectures, both old and new, are highlighted.

Computing a centerpoint of a finite planar set of points in linear time

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalization of the concept of the median of a set of reals. In this paper we present a linear-time algorithm for computing a centerpoint of a set ofn points in the plane, which is optimal compared with theO(n log3n) complexity of the previously best-known algorithm. We use suitable modifications of the hamsandwich cut algorithm in [Me2] and the prune-and-search technique of Megiddo [Me1] to achieve this improvement.

The maximum number of second smallest distances in finite planar sets

It is proved that the maximum numbers 2 of second smallest distances in anyn-element point set of the plane is less than 24/7n, where the constant 24/7 is best possible.

Fast algorithms for computing the diameter of a finite planar set

Three algorithms for computing the diameter of a finite planar set are proposed. Although all three algorithms have 0(n 2) worst-case running time, an expected-complexity analysis shows that, under a reasonable probability model, all three algorithms have linear expected running time. Experimental results indicate that two of these algorithms perform very well for some distributions, and are competitive with an existing method. Finally, we exhibit situations where exact algorithms out-perform an approximate algorithm.

The Number of Furthest Neighbour Pairs of a Finite Planar Set

Geometric problems involving a finite set of points in the plane have intrigued mathematicians for many years. As far back as 1893, J. J. Sylvester [4] posed the following problem: If S is a set of points in the plane, not all collinear, must there exist a line that contains exactly two points of S? This problem remained unsolved for forty years until it was rediscovered by P. Erdos in 1933 and solved a few days later by T. Gallai. This problem and a myriad of other famous problems in plane geometry are discussed in the fascinating paper by V. Klee [2]. In this note, we are concerned with results about maximum distances between pairs of points in the plane. An early problem of this type was posed by Hopf and Pannwitz [1] in 1934: What is the maximum number of pairs of points that can realize the diameter, the maximum distance between points of a set? The answer to this question appears in [3] and is the same as the number of points in the set. This is achieved by placing one point at the center of a circle of unit radius and the remaining points on a part of the boundary of the circle so that the two most distant points also are at distance 1 apart. We shall state and solve a variation of this problem. Consider a set N of n points in the Eucidean plane indexed by the integers 1, 2,.. ., n. A point j is called a furthest neighbour of a point i if d(i, j) = maxl,k n d(i,k), where d is the Euclidean distance function. Each point has at least one, and possibly several, furthest neighbours. Let n, denote the number of furthest neighbours of point i. Finally let

The Number of Furthest Neighbour Pairs of a Finite Planar Set

The Number of Furthest Neighbour Pairs of a Finite Planar Set

Optimal algorithms for computing the minimum distance between two finite planar sets

It is shown in this paper that the minimum distance between two finite planar sets of n points can be computer in O( n log n) worst-case running time and that this is optimal to within a constant factor. Furthermore, when the sets form a convex polygon this complexity can be reduced O( n).

Time- and storage-efficient implementation of an optimal planar convex hull algorithm

An implementation of an algorithm for computing the convex hull of a finite planar set of points is presented. The program is compared with an algorithm for the same purpose coded previously. Experimental results indicate that our program is superior to the other in terms of both running time and storage requirements.

Efficient algorithms for computing the maximum distance between two finite planar sets

An 0( n log n) algorithm is presented for computing the maximum euclidean distance between two finite planar sets of n points. When the n points form the vertices of simple polygons this complexity can be reduced to 0( n). The algorithm is empirically compared to the brute-force method as well as an alternate 0( n 2) algorithm. Both the 0( n log n) and 0( n 2) algorithms run in 0( n) expected time for many underlying distributions of the points. An ϵ-approximate algorithm can be obtained that runs in 0(n + 1 ϵ ) worst-case time.