Line Transversals in Large T (3)- and T (4)-Families of Congruent Discs.
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| Title: | Line Transversals in Large T (3)- and T (4)-Families of Congruent Discs. |
|---|---|
| Authors: | Aladar Heppes1 |
| Source: | Discrete & Computational Geometry. Sep2008, Vol. 40 Issue 2, p312-318. 7p. 7 Diagrams. |
| Subjects: | Line geometry, Geometric congruences, Diameter, Distance geometry, Discrete geometry, Differential geometry |
| Abstract: | Abstract A family of closed discs is said to have property T(k) if to every subset of at most k discs there belongs a common line transversal. A family of discs is said to be d-disjoint, d≥1, if the mutual distance between the centers of the discs is larger than d. It is known that a d-disjoint T(3)-family ℱ of unit diameter discs has a line transversal if . Similarly, a d-disjoint T(4)-family has a line transversal if . Both results are sharp in d, i.e., they do not hold for smaller values of d. The main result of this paper is that while the above lower bounds on d cannot be relaxed in general, some reduction of d can be compensated by imposing a proper d-dependent lower bound on the size of the family in both cases. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Line Transversals in Large T (3)- and T (4)-Families of Congruent Discs. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Aladar+Heppes%22">Aladar Heppes</searchLink><relatesTo>1</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Sep2008, Vol. 40 Issue 2, p312-318. 7p. 7 Diagrams. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Line+geometry%22">Line geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+congruences%22">Geometric congruences</searchLink><br /><searchLink fieldCode="DE" term="%22Diameter%22">Diameter</searchLink><br /><searchLink fieldCode="DE" term="%22Distance+geometry%22">Distance geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Discrete+geometry%22">Discrete geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+geometry%22">Differential geometry</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract  A family of closed discs is said to have property T(k) if to every subset of at most k discs there belongs a common line transversal. A family of discs is said to be d-disjoint, d≥1, if the mutual distance between the centers of the discs is larger than d. It is known that a d-disjoint T(3)-family ℱ of unit diameter discs has a line transversal if . Similarly, a d-disjoint T(4)-family has a line transversal if . Both results are sharp in d, i.e., they do not hold for smaller values of d. The main result of this paper is that while the above lower bounds on d cannot be relaxed in general, some reduction of d can be compensated by imposing a proper d-dependent lower bound on the size of the family in both cases. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-008-9071-0 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 7 StartPage: 312 Subjects: – SubjectFull: Line geometry Type: general – SubjectFull: Geometric congruences Type: general – SubjectFull: Diameter Type: general – SubjectFull: Distance geometry Type: general – SubjectFull: Discrete geometry Type: general – SubjectFull: Differential geometry Type: general Titles: – TitleFull: Line Transversals in Large T (3)- and T (4)-Families of Congruent Discs. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Aladar Heppes IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2008 Type: published Y: 2008 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 40 – Type: issue Value: 2 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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