scholarly journals On the complements of 3-dimensional convex polyhedra as polynomial images of ℝ3

2014 ◽  
Vol 25 (07) ◽  
pp. 1450071 ◽  
Author(s):  
José F. Fernando ◽  
Carlos Ueno
Keyword(s):  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Semialgebraic Sets ◽  
3 Dimensional

Let [Formula: see text] be a convex polyhedron of dimension n. Denote [Formula: see text] and let [Formula: see text] be its closure. We prove that for n = 3 the semialgebraic sets [Formula: see text] and [Formula: see text] are polynomial images of ℝ3. The former techniques cannot be extended in general to represent the semialgebraic sets [Formula: see text] and [Formula: see text] as polynomial images of ℝn if n ≥ 4.

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.

On the Topological Characterization of 3-D Polyhedral Microstrutures

2007 ◽  
Vol 537-538 ◽  
pp. 563-570 ◽  
Author(s):  
Tamás Réti ◽  
Agnes Csizmazia ◽  
Imre Felde
Keyword(s):  
Single Phase ◽  
Cellular Systems ◽  
Convex Polyhedra ◽  
Quantitative Characterization ◽  
Topological Characterization ◽  
Shape Factors ◽  
3 Dimensional ◽  
Cellular Models ◽  
Finite Set

To characterize topologically the polycrystalline microstructure of single-phase alloys computer simulations are performed on 3-dimensional cellular models. These infinite periodic cellular systems are constructed from a finite set of space filling convex polyhedra (grains). It is shown that the appropriately selected topological shape factors can be successfully used for the quantitative characterization of computer-simulated microstructures of various types.

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.

On a 3-Dimensional Isoperimetric Problem

1970 ◽  
Vol 13 (4) ◽  
pp. 447-449 ◽  
Author(s):  
Magelone Kömhoff
Keyword(s):  
Surface Area ◽  
Convex Polyhedron ◽  
Three Dimensional ◽  
Isoperimetric Problem ◽  
Total Surface Area ◽  
Edge Length ◽  
Regular Tetrahedron ◽  
3 Dimensional ◽  
Image Position

Let L(P) denote the total edge length and A(P) the total surface area of a three-dimensional convex polyhedron P. In [5] it was shown that if P belongs to the set of all polyhedra with triangular faces then for all with equality if and only if is a regular tetrahedron.It is not difficult to establish the inequality

scholarly journals An analogue of a theorem of Steinitz for ball polyhedra in $$\mathbb {R}^3$$

2021 ◽  
Author(s):  
Sami Mezal Almohammad ◽  
Zsolt Lángi ◽  
Márton Naszódi
Keyword(s):  
Convex Polyhedron ◽  
3 Dimensional ◽  
Edge Graph ◽  
Steinitz’S Theorem

AbstractSteinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $$\mathbb {R}^3$$ R 3 .

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.

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