Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles.
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| Title: | Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles. |
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| Authors: | Cho, Yunhi1,2 (AUTHOR) yhcho@uos.ac.kr, Kim, Seonhwa1 (AUTHOR) seonhwa17kim@uos.ac.kr |
| Source: | Discrete & Computational Geometry. Sep2025, Vol. 74 Issue 2, p302-336. 35p. |
| Subjects: | Dihedral angles, Polyhedra, Hypothesis, Geometric rigidity, Geometry |
| Abstract: | We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and (ii) it does not have partially-flat vertices, and under an additional technical requirement that (iii) any triple of vertices is not collinear. The proof is consistently valid for Euclidean, hyperbolic and spherical geometry, which takes a completely different approach from the argument of the Cauchy rigidity theorem. Various counterexamples are provided that arise when these conditions are violated, and self-contained proofs are presented whenever possible. As a corollary, the rigidity of several families of polyhedra is also established. Finally, we propose two conjectures: the first suggests that Condition (iii) can be removed, and the second concerns the rigidity of spherical nonconvex polygons. [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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| Header | DbId: egs DbLabel: Engineering Source An: 188126545 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Cho%2C+Yunhi%22">Cho, Yunhi</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> yhcho@uos.ac.kr</i><br /><searchLink fieldCode="AR" term="%22Kim%2C+Seonhwa%22">Kim, Seonhwa</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> seonhwa17kim@uos.ac.kr</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Sep2025, Vol. 74 Issue 2, p302-336. 35p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Dihedral+angles%22">Dihedral angles</searchLink><br /><searchLink fieldCode="DE" term="%22Polyhedra%22">Polyhedra</searchLink><br /><searchLink fieldCode="DE" term="%22Hypothesis%22">Hypothesis</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+rigidity%22">Geometric rigidity</searchLink><br /><searchLink fieldCode="DE" term="%22Geometry%22">Geometry</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and (ii) it does not have partially-flat vertices, and under an additional technical requirement that (iii) any triple of vertices is not collinear. The proof is consistently valid for Euclidean, hyperbolic and spherical geometry, which takes a completely different approach from the argument of the Cauchy rigidity theorem. Various counterexamples are provided that arise when these conditions are violated, and self-contained proofs are presented whenever possible. As a corollary, the rigidity of several families of polyhedra is also established. Finally, we propose two conjectures: the first suggests that Condition (iii) can be removed, and the second concerns the rigidity of spherical nonconvex polygons. [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/s00454-024-00670-w Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 35 StartPage: 302 Subjects: – SubjectFull: Dihedral angles Type: general – SubjectFull: Polyhedra Type: general – SubjectFull: Hypothesis Type: general – SubjectFull: Geometric rigidity Type: general – SubjectFull: Geometry Type: general Titles: – TitleFull: Rigidity of Nonconvex Polyhedra with Respect to Edge Lengths and Dihedral Angles. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Cho, Yunhi – PersonEntity: Name: NameFull: Kim, Seonhwa IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 74 – Type: issue Value: 2 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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