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Rigidity Theory of Circles, Polygons, and Polyhedra
Rigidity Theory of Circles, Polygons, and Polyhedra
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211151347
- ISBN
- 9798382843247
- DDC
- 510
- 저자명
- Zhang, Zhen.
- 서명/저자
- Rigidity Theory of Circles, Polygons, and Polyhedra
- 발행사항
- [Sl] : Cornell University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 127 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
- 주기사항
- Advisor: Connelly, Robert.
- 학위논문주기
- Thesis (Ph.D.)--Cornell University, 2024.
- 초록/해제
- 요약Rigidity theory is the study of the uniqueness of structures. In most cases, a structure is defined by a set of variables with constraints, ideally polynomial, that come from geometry. A constraint can be an equality or an inequality. There are various types of rigidity. Intuitively, local rigidity means uniqueness in a small neighborhood, global rigidity means uniqueness in a much larger space at the given dimension, and universal rigidity means uniqueness in all higher dimensions.This thesis explores the rigidity of several common structures that are slightly more complex than a set of points with fixed distance constraints, known as the bar-joint frameworks. These structures include circle packings, polyhedra, and various special sets of points with inequality constraints, known as tensegrities. They are natural extensions of the bar-joint frameworks that occur in many studies. The rigidity of circles with a given tangency pattern and fixed radii has been well studied. This is known as sticky disks, as the disks that are required to be tangent must "stick together''. It is natural to ask if the rigidity results for sticky disks still hold with flexible radii. Another problem arises from the study of polytopes as regards the rigidity of a polyhedron with fixed edge lengths and vertices of each facet staying in the same plane. To solve these problems, we extend the methods used for bar-joint frameworks so that the algebra behaves analogously. Several examples of structures with interesting results on rigidity are given in each chapter. Often, rigidity is done in a "generic'' sense where singularities are ignored. An ambitious attempt is made to replace the assumption "generic'' with the assumption "convex'' for several classes of bar-joint frameworks. Some examples of resolved cases and an open case are given in the last chapter.
- 일반주제명
- Mathematics
- 일반주제명
- Computer science
- 일반주제명
- Theoretical mathematics
- 키워드
- Circle packings
- 키워드
- Combinatorics
- 키워드
- Convex geometry
- 키워드
- Polyhedra
- 키워드
- Rigidity
- 기타저자
- Cornell University Applied Mathematics
- 기본자료저록
- Dissertations Abstracts International. 85-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■00520250211151347
■006m o d
■007cr#unu||||||||
■020 ▼a9798382843247
■035 ▼a(MiAaPQ)AAI31242707
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aZhang, Zhen.▼0(orcid)0000-0001-9789-3674
■24510▼aRigidity Theory of Circles, Polygons, and Polyhedra
■260 ▼a[Sl]▼bCornell University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a127 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 85-12, Section: B.
■500 ▼aAdvisor: Connelly, Robert.
■5021 ▼aThesis (Ph.D.)--Cornell University, 2024.
■520 ▼aRigidity theory is the study of the uniqueness of structures. In most cases, a structure is defined by a set of variables with constraints, ideally polynomial, that come from geometry. A constraint can be an equality or an inequality. There are various types of rigidity. Intuitively, local rigidity means uniqueness in a small neighborhood, global rigidity means uniqueness in a much larger space at the given dimension, and universal rigidity means uniqueness in all higher dimensions.This thesis explores the rigidity of several common structures that are slightly more complex than a set of points with fixed distance constraints, known as the bar-joint frameworks. These structures include circle packings, polyhedra, and various special sets of points with inequality constraints, known as tensegrities. They are natural extensions of the bar-joint frameworks that occur in many studies. The rigidity of circles with a given tangency pattern and fixed radii has been well studied. This is known as sticky disks, as the disks that are required to be tangent must "stick together''. It is natural to ask if the rigidity results for sticky disks still hold with flexible radii. Another problem arises from the study of polytopes as regards the rigidity of a polyhedron with fixed edge lengths and vertices of each facet staying in the same plane. To solve these problems, we extend the methods used for bar-joint frameworks so that the algebra behaves analogously. Several examples of structures with interesting results on rigidity are given in each chapter. Often, rigidity is done in a "generic'' sense where singularities are ignored. An ambitious attempt is made to replace the assumption "generic'' with the assumption "convex'' for several classes of bar-joint frameworks. Some examples of resolved cases and an open case are given in the last chapter.
■590 ▼aSchool code: 0058.
■650 4▼aMathematics
■650 4▼aComputer science
■650 4▼aTheoretical mathematics
■653 ▼aCircle packings
■653 ▼aCombinatorics
■653 ▼aConvex geometry
■653 ▼aDistance geometry
■653 ▼aPolyhedra
■653 ▼aRigidity
■690 ▼a0405
■690 ▼a0984
■690 ▼a0642
■71020▼aCornell University▼bApplied Mathematics.
■7730 ▼tDissertations Abstracts International▼g85-12B.
■790 ▼a0058
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17161366▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
