On Lines and Joints.
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| Title: | On Lines and Joints. |
|---|---|
| Authors: | Kaplan, Haim1 haimk@post.tau.ac.il, Sharir, Micha michas@post.tau.ac.il, Shustin, Eugenii2 shustin@post.tau.ac.il |
| Source: | Discrete & Computational Geometry. Dec2010, Vol. 44 Issue 4, p838-843. 6p. |
| Subjects: | Line geometry, Algebraic curves, Mathematical proofs, Number theory, Boundary value problems, Dimensional analysis, Mathematical analysis |
| Abstract: | Let L be a set of n lines in ℝ, for d≥3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ( n). For d=3, this is a considerable simplification of the original algebraic proof of Guth and Katz (Algebraic methods in discrete analogs of the Kakeya problem, 4 December , ), and of the follow-up simpler proof of Elekes et al. (On lines, joints, and incidences in three dimensions. Manuscript, 11 May , ). Some extensions, e.g., to the case of joints of algebraic curves, are also presented. [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: On Lines and Joints. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Kaplan%2C+Haim%22">Kaplan, Haim</searchLink><relatesTo>1</relatesTo><i> haimk@post.tau.ac.il</i><br /><searchLink fieldCode="AR" term="%22Sharir%2C+Micha%22">Sharir, Micha</searchLink><i> michas@post.tau.ac.il</i><br /><searchLink fieldCode="AR" term="%22Shustin%2C+Eugenii%22">Shustin, Eugenii</searchLink><relatesTo>2</relatesTo><i> shustin@post.tau.ac.il</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Dec2010, Vol. 44 Issue 4, p838-843. 6p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Line+geometry%22">Line geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+curves%22">Algebraic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+proofs%22">Mathematical proofs</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Dimensional+analysis%22">Dimensional analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Let L be a set of n lines in ℝ, for d≥3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ( n). For d=3, this is a considerable simplification of the original algebraic proof of Guth and Katz (Algebraic methods in discrete analogs of the Kakeya problem, 4 December , ), and of the follow-up simpler proof of Elekes et al. (On lines, joints, and incidences in three dimensions. Manuscript, 11 May , ). Some extensions, e.g., to the case of joints of algebraic curves, are also presented. [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/s00454-010-9246-3 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 6 StartPage: 838 Subjects: – SubjectFull: Line geometry Type: general – SubjectFull: Algebraic curves Type: general – SubjectFull: Mathematical proofs Type: general – SubjectFull: Number theory Type: general – SubjectFull: Boundary value problems Type: general – SubjectFull: Dimensional analysis Type: general – SubjectFull: Mathematical analysis Type: general Titles: – TitleFull: On Lines and Joints. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Kaplan, Haim – PersonEntity: Name: NameFull: Sharir, Micha – PersonEntity: Name: NameFull: Shustin, Eugenii IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2010 Type: published Y: 2010 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 44 – Type: issue Value: 4 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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