We are given a read-only memory for input and a write-only stream for output. For a positive integer parameter s, an s-workspace algorithm is an algorithm using only O(s) words of workspace in addition to the memory for input. In this paper, we present an O(n/s)-time s-workspace algorithm for subdividing a simple polygon into O(min{n/s, s}) subpolygons of complexity O(max{n/s, s}). As applicati...

Bicycle (n, k)-gons are equilateral n-gons whose k-diagonals are equal. In this paper, the order of infinitesimal flexibility of the regular n-gon within the family of bicycle (n, k)-gons is studied. An equation characterizing first order flexible regular bicycle (n, k)-gons were computed by S. Tabachnikov in [7]. This equation was solved by R. Connelly and the author in [3]. S. Tabachnikov has...

Introduction Let P = p1, . . . , pn be a simple polygon (all polygons are assumed convex throughout this paper). A fundamental problem in geometric optimization is to compute a minimum-area or a minimum-perimeter convex k-gon (denoted QA or Qp, resp.) that encloses P . While efficient algorithms for finding QA are known for more than 20 years [8, 1, 2], the problem of finding Qp has remained op...

A small polygon is a of unit diameter. The maximal area with $$n=2m$$ vertices not known when $$m\ge 7$$ . In this paper, we construct, for each and 3$$ , n-gon whose the value one-variable function. We show that, all even $$n\ge 6$$ obtained improves by $$O(1/n^5)$$ that best prior constructed Mossinghoff. particular, $$n=6$$ 6-gon has area.

A finite generalized 2d-gon of order (s, t) with d ∈ {2, 3, 4} and s 6= 1 is called extremal if t attains its maximal possible value sed , where e2 = e4 = 2 and e3 = 3. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized 2d-gon of order (s, t) to be extremal has so far only be solved for quadrangles and octagons. In this paper, we obta...

Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n logn + o(n logn) and the expected number of empty convex four-gons with vertices from S is Θ(n). keywords: random point set, e...

In a previous paper (given at the International Congress of Mathematicians at Nice 1970) entitled "On the application of combinatorial analysis to number theory geometry and analysis", I discussed many combinatorial results and their applications . I will refer to this paper as I (as much as possible I will try to avoid overlap with this paper) . First I discuss those problems mentioned in I wh...

The Hamiltonian potentials of the bending deformations of n-gons in E 3 studied in KM] and Kly] give rise to a Hamiltonian action of the Malcev Lie algebra P n of the pure braid group P n on the moduli space M r of n-gon linkages with the side-lengths r = (r 1 ; :::; r n) in E 3. If e 2 M r is a singular point we may linearize the vector elds in P n at e. This linearization yields a at connecti...

For a class of potential functions including those used for the planar n-body and n-vortex problems, we investigate co-circular central configurations whose center of mass coincides with the center of the circle containing the bodies. Useful equations are derived that completely describe the problem. Using a topological approach, it is shown that for any choice of positive masses (or circulatio...

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach [2, Problem 3, p. 437] asked the following question: For n ≥ 5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine [1] answered this question, and showed that t...