Maximal Areas of Reuleaux Polygons

1970 ◽  
Vol 13 (2) ◽  
pp. 175-179 ◽  
Author(s):  
G. T. Sallee
Keyword(s):  
Finite Number ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Circular Arcs ◽  
Positive Length ◽  
Reuleaux Polygon ◽  
Reuleaux Polygons

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

scholarly journals Diameter, width and thickness in the hyperbolic plane

2021 ◽  
Vol 112 (3) ◽  
Author(s):  
Ákos G. Horváth
Keyword(s):  
Convex Body ◽  
Hyperbolic Geometry ◽  
Hyperbolic Plane ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Plane Convex Body ◽  
Constant Diameter

AbstractIn hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.

Curves on Surfaces of Constant Width

1966 ◽  
Vol 9 (1) ◽  
pp. 15-22
Author(s):  
William W. Armstrong
Keyword(s):  
Convex Set ◽  
Dimensional Space ◽  
Constant Width ◽  
Smoothness Condition ◽  
3 Dimensional ◽  
Curves On Surfaces ◽  
Class C

A surface S of constant width is the boundary of a convex set K of constant width in euclidean 3-dimensional space E3. (See [l] pp. 127–139. )Our first result concerns the interdependence of five properties which a curve on such a surface may possess. Let S be a surface of constant width D > 0 which satisfies the smoothness condition that it be a 2-dimensional submanifold of E3 of class C2.

Applications of outwardly simple line families to plane convex sets

1977 ◽  
Vol 82 (3) ◽  
pp. 353-356 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Plane Convex ◽  
Minimal Width ◽  
Parallel Lines

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

Stability Theorems for Convex Domains of Constant Width

1988 ◽  
Vol 31 (3) ◽  
pp. 328-337 ◽  
Author(s):  
H. Groemer
Keyword(s):  
Convex Domain ◽  
Hausdorff Distance ◽  
Stability Problem ◽  
Constant Width ◽  
Circular Disc ◽  
Convex Domains ◽  
Plane Convex ◽  
Reuleaux Triangle ◽  
Stability Theorems ◽  
Maximal Area

AbstractIt is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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