scholarly journals On Polyhedral Realization with Isosceles Triangles

2021 ◽  
Author(s):  
David Eppstein
Keyword(s):  
Convex Polyhedron ◽  
Infinite Family ◽  
Convex Polyhedra ◽  
Dominating Sets ◽  
Polyhedral Graph ◽  
Bounded Size ◽  
Scalene Triangle ◽  
Isosceles Triangles

AbstractAnswering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer i such that all convex i-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (Renaissance Banff, Bridges Conference, pp. 489–496, 2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.

scholarly journals Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021 ◽  
Vol 2 ◽  
Author(s):  
Zhiping Qiu ◽  
Han Wu ◽  
Isaac Elishakoff ◽  
Dongliang Liu
Keyword(s):  
Linear Programming ◽  
Convex Polyhedron ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Composite Plates ◽  
Convex Polyhedra ◽  
Chebyshev Inequality ◽  
Engineering Structures ◽  
Load Bearing ◽  
Polyhedron Model

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

Approximating Convex Polyhedra with Axis-Parallel Boxes

1997 ◽  
Vol 07 (03) ◽  
pp. 253-267 ◽  
Author(s):  
Binhai Zhu
Keyword(s):  
Hausdorff Distance ◽  
Convex Polyhedron ◽  
Linear Time ◽  
Time Algorithm ◽  
Convex Polyhedra ◽  
Linear Time Algorithm ◽  
Brute Force ◽  
Axis Parallel ◽  
The Given ◽  
Fixed Center

In this paper, we present an O(n4 log 2n) time algorithm to compute an approximate discrete axis-parallel box of a given n-vertex convex polyhedron P such that the given polyhedron is minimized. Here, "discrete" means that each plane containing a face of the approximate box passes through a vertex of P (or, more generally, passes through a point of a set of given points). This algorithm is significantly faster than the brute force O(n7) time solution for computing the optimal approximate axis-parallel box A* of P such that the symmetric difference of the volume between P and A* is minimized. We present a linear time algorithm to compute a pseudo-optimal (with factor [Formula: see text] approximate axis-parallel box of a convex polyhedron under the Hausdorff distance criterion. We also present O(n) and O(n7 log n) time algorithms to compute the optimal approximate ball, with or without a fixed center, of a convex polyhedron under the Hausdorff distance criterion.

COMPUTING NICE PROJECTIONS OF CONVEX POLYHEDRA

2011 ◽  
Vol 21 (01) ◽  
pp. 71-85
Author(s):  
MD. ASHRAFUL ALAM ◽  
MASUD HASAN
Keyword(s):  
Orthogonal Projection ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Projected Area

In an orthogonal projection of a convex polyhedron P, the visibility ratio of a face f (of an edge e) is the ratio of orthogonally projected area of f (length of e) and its actual area (length). In this paper, we give algorithms for nice projections of P such that the minimum visibility ratio among all visible faces (among all visible edges) is maximized.

The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra

1963 ◽  
Vol 15 ◽  
pp. 744-751 ◽  
Author(s):  
B. Grünbaum ◽  
T. S. Motzkin
Keyword(s):  
Euclidean Space ◽  
Convex Polyhedron ◽  
Three Dimensional ◽  
Convex Polyhedra ◽  
Dimensional Euclidean Space

The problem of determining the possible morphological types of convex polyhedra in three-dimensional Euclidean space E3 is well known to be quite hopeless. We lack not only any general way of determining whether there exists a convex polyhedron having as faces ƒ3 triangles, ƒ4 quadrangles, . . . , and ƒnn-gons, but even much more special questions of this kind seem to be rather elusive.

Wright-Fisher Geometry

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Normal Form ◽  
Euclidean Space ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Linearly Independent ◽  
Regular Convex ◽  
Normal Vectors ◽  
Manifolds With Corners ◽  
Natural Domains ◽  
Manifold With Corners

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural domains of definition for generalized Kimura diffusions: polyhedra in Euclidean space or, more generally, abstract manifolds with corners. Amongst the convex polyhedra, the chapter distinguishes the subclass of regular convex polyhedra P. P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors that are linearly independent. These definitions establish that any regular convex polyhedron is a manifold with corners. The chapter concludes by defining the general class of elliptic Kimura operators on a manifold with corners P and shows that there is a local normal form for any operator L in this class.

Collapsing three-dimensional convex polyhedra: correction

1980 ◽  
Vol 88 (2) ◽  
pp. 307-310 ◽  
Author(s):  
D. R. J. Chillingworth
Keyword(s):  
Convex Polyhedron ◽  
Three Dimensional ◽  
Convex Polyhedra

An ingenious construction due to Connelly and Henderson (2) has shown that there exists a rectilinearly triangulated convex polyhedron P in having the property that at least one vertex of the triangulation lies in the interior of a face of P, and yet there is no isomorphic triangulation of a convex polyhedron P′ all of whose vertices are vertices of P′. Thus the assertion beginning on the top line of p. 354 of (1) is false, which leaves a gap in the proof of essentially the main result of (1), namely that any rectilinearly triangulated convex polyhedron incan be simplicially collapsed onto its boundary minus a 2-simplex σ. The purpose of this note is to show that the theorem is nevertheless still true. In any case the Corollaries 2 and 3 in (1) are unaffected by the error.

Ordering of convex polyhedra and the Fedorov algorithm

2017 ◽  
Vol 73 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Yury L. Voytekhovsky
Keyword(s):  
Adjacency Matrix ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Numerical Code ◽  
Edge Graph ◽  
The Relationship

A method of naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph has been previously suggested. A polyhedron can be built using its name. Classes of convexn-acra (i.e.n-vertex polyhedra) are strictly (without overlapping) ordered by their names. In this paper the relationship between the Fedorov algorithm to generate the whole combinatorial variety of convex polyhedra and the above ordering is described. The convexn-acra are weakly ordered by the maximum extra valencies of their vertices. Thus, non-simplen-acra follow the simple ones for anyn.

Some properties of Kiepert lines of a triangle

2016 ◽  
Vol 100 (547) ◽  
pp. 54-67
Author(s):  
Michael Fox
Keyword(s):  
Single Point ◽  
Common Base ◽  
The Common ◽  
Scalene Triangle ◽  
Isosceles Triangles

This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.

How to name and order convex polyhedra

2016 ◽  
Vol 72 (5) ◽  
pp. 582-585 ◽  
Author(s):  
Yury L. Voytekhovsky
Keyword(s):  
Adjacency Matrix ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Numerical Code ◽  
Edge Graph

In this paper a method is suggested for naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph. A polyhedron is uniquely fixed by its name and can be built using it. Classes of convexn-acra (i.e.n-vertex polyhedra) are strictly ordered by their names.

The Correspondence Between Convex Polydedra and Tensegrity Systems: A Classification System

1992 ◽  
Vol 7 (2) ◽  
pp. 115-125 ◽  
Author(s):  
Robert Grip
Keyword(s):  
Classification System ◽  
Convex Polyhedron ◽  
Original System ◽  
Convex Polyhedra ◽  
Space Structures ◽  
Space Filling ◽  
Space Frame ◽  
Tensegrity Structure ◽  
Tensegrity Systems ◽  
Geometric Duality

Since tensegrity systems are often topologically and geometrically complex it is important to have a simple and effective method to describe and classify them in an orderly manner. This paper sets forth a system of relating any convex polyhedron to a tensegrity by defining a one-to-one correspondence between the elements of the polyhedron and the elements of the tensegrity. This system, which was first introduced by the author in a 1978 booklet while working in Buckminster Fuller's Philadelphia office, is now extended to include agglomerations of convex polyhedra — both regular and irregular space-filling arrays. In both the original system and its extension, the principle of geometric duality is clearly demonstrated. More importantly for engineers, the method will allow those familiar with space structures an easy and convenient way to relate any space frame or truss to a corresponding tensegrity structure.

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