scholarly journals Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

2021 ◽  
Author(s):  
Ferenc Fodor
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Plane Convex ◽  
Convex Disc ◽  
Convex Curves

AbstractEggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

scholarly journals On the relative distances of six points in a plane convex body

2005 ◽  
Vol 42 (3) ◽  
pp. 253-264
Author(s):  
Károly Böröczky ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Relative Distance ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

Frieze Patterns in the Hyperbolic Plane

1974 ◽  
Vol 17 (1) ◽  
pp. 45-50 ◽  
Author(s):  
C. W. L. Garner
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Symmetry Groups ◽  
Euclidean Case ◽  
Parallel Lines

AbstractIt is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

Regular compound tessellations of the hyperbolic plane

1964 ◽  
Vol 278 (1373) ◽  
pp. 147-167 ◽  
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Regular Polyhedron ◽  
Alternating Group ◽  
Platonic Solids ◽  
Icosahedral Group ◽  
Analogous Compound ◽  
Equilateral Triangles ◽  
Two Parameter

Kepler, in his Harmonice mundi of 1619, extended the idea of a regular polyhedron in at least two directions. Observing that two equal regular tetrahedra can interpenetrate in such a way that their twelve edges are the diagonals of the six faces of a cube, he called this combination stella octangida . It is occasionally found in nature as twinned crystals of tetrahedrite, Cu 10 (Zn, Fe, Cu) 2 Sb 4 S 13 . In addition to this ‘compound’ of two tetrahedra inscribed in a cube, there are several other compound polyhedra, the prettiest being the compound of five tetrahedra inscribed in a dodecahedron. The icosahedral group of rotations may be described as the alternating group on these five tetrahedra. Kepler observed also that the tessellation of squares (or regular hexagons, or equilateral triangles), filling and covering the Euclidean plane, may be regarded as an infinite analogue of the spherical tessellations which are ‘blown-up’ versions of the Platonic solids. Putting these two ideas together, one naturally regards the compound polyhedra as compound tessellations of the sphere. The analogous compound tessellations of the Euclidean plane (18 two-parameter families of them) were enumerated in 1948. The present paper describes many compound tessellations of the hyperbolic plane: five one-parameter families and seventeen isolated cases. It is conjectured that this list is complete, but there remains the possibility that a few more isolated cases may still be discovered.

Touching Convex Sets in the Plane

1994 ◽  
Vol 37 (4) ◽  
pp. 495-504 ◽  
Author(s):  
Meir Katchalski ◽  
János Pach
Keyword(s):  
Positive Constant ◽  
Euclidean Plane ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
The Other ◽  
Plane Convex ◽  
Straight Line

AbstractTwo subsets of the Euclidean plane touch each other if they have a point in common and there is a straight line separating one from the other.It is shown that there exists a positive constant c such that if are families of plane convex sets with for some k ≥ 1 and if every touches every then either contains k members having nonempty intersection.

Minimal properties for convex annuli of plane convex curves

1995 ◽  
Vol 64 (3) ◽  
pp. 254-263 ◽  
Author(s):  
C. Peri ◽  
S. Vassallo
Keyword(s):  
Plane Convex ◽  
Convex Curves

An Analogue of Napoleon’s Theorem in the Hyperbolic Plane

2001 ◽  
Vol 44 (3) ◽  
pp. 292-297 ◽  
Author(s):  
Angela McKay
Keyword(s):  
Equilateral Triangle ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Equilateral Triangles ◽  
Napoleon’S Theorem

AbstractThere is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane.

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