A Theorem on a Cyclic Polygon

1940 ◽  
Vol 47 (7) ◽  
pp. 466
Author(s):  
R. Goormaghtigh
Keyword(s):  
Cyclic Polygon

Self assembly of cyclic polygon shaped fluid colloidal membranes through pinning

Soft Matter ◽  
2018 ◽  
Vol 14 (48) ◽  
pp. 9959-9966
Author(s):  
Lachit Saikia ◽  
Prerna Sharma
Keyword(s):  
Phase Separation ◽  
Self Assembly ◽  
Anisotropic Fluid ◽  
Binary Phase ◽  
Colloidal Membranes ◽  
Cyclic Polygon ◽  
Colloidal Rods

Binary phase separation of geometrically different colloidal rods in the presence of pinning leads to cyclic polygon shaped anisotropic fluid colloidal membranes.

scholarly journals Equivalent cyclic polygon of a euclidean travelling salesman problem tour and modified formulation

2021 ◽  
Author(s):  
A. Herraiz ◽  
M. Gutierrez ◽  
M. Ortega-Mier
Keyword(s):  
Objective Function ◽  
Travelling Salesman Problem ◽  
Geometric Parameters ◽  
Geometric Transformation ◽  
Travelling Salesman ◽  
New Formulation ◽  
Additional Constraints ◽  
Cyclic Polygon ◽  
The Way

AbstractWe define a geometric transformation of Euclidean Travelling Salesman Problem (TSP) tours that leads to a new formulation of the TSP. For every Euclidean TSP n-city tour, it is possible to construct an inscribed n-polygon (Equivalent Cyclic Polygon, ECP) such that the lengths of the edges are equal to the corresponding TSP tour links and follow the same sequence order. The analysis of the ECP elicits the possibility of defining a new objective function in terms of angles instead of distances. This modification opens the way to identify characterizing geometric parameters of the TSP as well as to explore new heuristics based on the inclusion of additional constraints. The experimentation with a set of cases shows promising results compared to the traditional compact formulations. The behavior of the ECP-based TSP formulations is better when the nodes of the TSP are randomly or evenly distributed.

scholarly journals Critical configurations of planar robot arms

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Giorgi Khimshiashvili ◽  
Gaiane Panina ◽  
Dirk Siersma ◽  
Alena Zhukova
Keyword(s):  
Critical Points ◽  
Morse Index ◽  
Robot Arms ◽  
Polygonal Chain ◽  
Oriented Area ◽  
Critical Configurations ◽  
Planar Robot ◽  
Polygonal Chains ◽  
Closed Polygon ◽  
Cyclic Polygon

AbstractIt is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

A Theorem on a Cyclic Polygon

1940 ◽  
Vol 47 (7) ◽  
pp. 466-468
Author(s):  
R. Goormaghtigh
Keyword(s):  
Cyclic Polygon

scholarly journals Generalization of Heron’s and Brahmagupta’s equalities to any cyclic polygon

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Enrico Laeng
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Explicit Solutions ◽  
Elementary Approach ◽  
The Real ◽  
Simple Geometry ◽  
Special Cases ◽  
Cyclic Polygon

AbstractIt is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having $$n>4$$ n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

scholarly journals Morse index of a cyclic polygon

2011 ◽  
Vol 9 (2) ◽  
pp. 364-377 ◽  
Author(s):  
Gaiane Panina ◽  
Alena Zhukova
Keyword(s):  
Morse Index ◽  
Cyclic Polygon

Nonholonomic control of distance-based cyclic polygon formation

2013 ◽  
Author(s):  
Byung-Hun Lee ◽  
Seung-Ju Lee ◽  
Myoung-Chul Park ◽  
Kwang-Kyo Oh ◽  
Hyo-Sung Ahn
Keyword(s):  
Nonholonomic Control ◽  
Cyclic Polygon

Control of inter-agent distances in cyclic polygon formations

2012 ◽  
Author(s):  
Myoung-Chul Park ◽  
Byeong-Yeon Kim ◽  
Kwang-Kyo Oh ◽  
Hyo-Sung Ahn
Keyword(s):  
Cyclic Polygon
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