Placing the largest similar copy of a convex polygon among polygonal obstacles

In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

Download Full-text

AN APPROXIMATION ALGORITHM FOR LOCATING MAXIMAL DISKS WITHIN CONVEX POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195911003858 ◽

2011 ◽

Vol 21 (06) ◽

pp. 661-684

Author(s):

HIROFUMI AOTA ◽

TAKURO FUKUNAGA ◽

HIROSHI NAGAMOCHI

Keyword(s):

Approximation Algorithm ◽

Convex Polygon ◽

Computation Time ◽

Approximation Ratio ◽

Convex Polygons ◽

The Given

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners.

Download Full-text

UNIQUENESS PROPERTIES OF THE INVARIATOR, LEADING TO SIMPLE COMPUTATIONS

Image Analysis & Stereology ◽

10.5566/ias.v31.p89-98 ◽

2012 ◽

Vol 31 (2) ◽

pp. 89 ◽

Cited By ~ 5

Author(s):

Luis Manuel Cruz-Orive

Keyword(s):

Convex Polygon ◽

Smooth Boundary ◽

Three Dimensional ◽

Computational Formula ◽

Compact Domain ◽

Particle Volume ◽

Pivotal Point ◽

Wedge Volume ◽

The Mean ◽

The One

It is shown that, for a three dimensional particle (namely an arbitrary compact domain with piecewise smooth boundary in R^3) the mean wedge volume defined on a given pivotal section is equal to the average nucleator estimator of the particle volume defined on that section. Further, if the particle is convex and it contains the pivotal point, then the flower area of a given pivotal section equals the average surfactor estimator defined on that section. These results are intended to throw some light on the standing conjecture that the functional defined on a pivotal section according to the invariator has a unique general expression. As a plus, the former result leads to a computational formula for the mean wedge volume of a convex polygon which is much simpler than the one published recently, and it is valid whether the fixed pivotal point is interior or exterior to the particle.

Download Full-text

An extremum problem for the power moment of a convex polygon contained in a disc

Advances in Geometry ◽

10.1515/advgeom-2021-0021 ◽

2021 ◽

Vol 21 (4) ◽

pp. 599-609

Author(s):

Irmina Herburt ◽

Shigehiro Sakata

Keyword(s):

Convex Polygon ◽

Extremum Problem ◽

Classical Theorem ◽

Power Moment ◽

Area Functional ◽

The Mean

Abstract In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.

Download Full-text