Continuous Families of Curves

1966 ◽  
Vol 18 ◽  
pp. 529-537 ◽  
Author(s):  
Branko Grünbaum
Keyword(s):  
Fixed Point ◽  
Jordan Curve ◽  
Convex Sets ◽  
General Setting ◽  
Jordan Curve Theorem ◽  
Planar Line ◽  
Families Of Curves ◽  
Planar Convex ◽  
Free Involution

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

On Planar Continuous Families of Curves

1969 ◽  
Vol 21 ◽  
pp. 513-530 ◽  
Author(s):  
Tudor Zamfirescu
Keyword(s):  
Convex Body ◽  
General Setting ◽  
The Other ◽  
Continuous Family ◽  
Families Of Curves ◽  
Family Of Curves ◽  
Planar Convex

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

scholarly journals Maximizing the overlap of two planar convex sets under rigid motions

2007 ◽  
Vol 37 (1) ◽  
pp. 3-15 ◽  
Author(s):  
Hee-Kap Ahn ◽  
Otfried Cheong ◽  
Chong-Dae Park ◽  
Chan-Su Shin ◽  
Antoine Vigneron
Keyword(s):  
Convex Sets ◽  
Planar Convex ◽  
Rigid Motions

scholarly journals Approximate inverse limits and (m,n)-dimensions

2021 ◽  
Vol 56 (1) ◽  
pp. 175-194
Author(s):  
James F. Peters ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Free Abelian Group ◽  
Betti Numbers ◽  
Boundary Region ◽  
Inverse Limits ◽  
Jordan Curve Theorem ◽  
Group Representations ◽  
Approximate Inverse ◽  
Fixed Cell

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.

scholarly journals A Proof of the Jordan Curve Theorem

2019 ◽  
Vol 11 (12) ◽  
pp. 345-360
Author(s):  
Xing Zhang
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem

Translations of planar convex sets

1971 ◽  
Vol 22 (1) ◽  
pp. 103-105 ◽  
Author(s):  
Togo Nishiura ◽  
Franz Schnitzer
Keyword(s):  
Convex Sets ◽  
Planar Convex

Walk-set induced connectedness in digital spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 247-256
Author(s):  
JOSEF SLAPAL ◽  
Keyword(s):  
Jordan Curve ◽  
Digital Images ◽  
Simple Graph ◽  
Jordan Curve Theorem ◽  
Strong Product ◽  
Digital Plane ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Digital Spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set Z2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on Z2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane Z2 for the study of digital images.

Sign in / Sign up

Export Citation Format

Share Document