東華大學圖書館 |

Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry./
作者:	Graham, Opal.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
面頁冊數:	94 p.
附註:	Source: Dissertations Abstracts International, Volume: 82-05, Section: B.
Contained By:	Dissertations Abstracts International82-05B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28002480
ISBN:	9798678122179

Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry.
Graham, Opal.

Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 94 p.

Source: Dissertations Abstracts International, Volume: 82-05, Section: B.

Thesis (Ph.D.)--The Florida State University, 2020.

This item must not be sold to any third party vendors.

This dissertation examines the study of rigidity of collections of objects in various geometric spaces, and the correspondences shared between geometries. In particular, we take a look at vectors and lines in Lorentz $(n+1)$-space, points, ideal points and hyperplanes in hyperbolic $n$-space, and circles and points in the Riemann sphere. One main objective is to explore how much information invariant to a given space is sufficient for a collection to be unique up to the transformations of the space. The answer to this question changes with the qualities a collection of objects possesses. To this end, this dissertation focuses on the role independence of objects plays in uniqueness. As another primary focus, a new invariant is introduced in each geometric setting to provide a means with which to study the rigidity of intermingled collections of objects that are infinitely far away from one another.The first chapter gives a history of circle, sphere, and point configurations, and the correspondences between configurations of objects in hyperbolic space and Lorentz space. All theorems stated in this chapter are well-established results and provide both motivation for studying conditions for rigidity and for introducing an invariant that allows for intermingled collections of points and spheres.In the second chapter, the main result is established entirely within the context of Lorentz space. An invariant of Lorentz transformations called the Lorentz ratio is defined that allows one to work with intermingled collections of vectors and light-like lines in Lorentz space. Three main statements are made about the rigidity of intermingled collections of vectors and light-like lines, each relying on a linearly independent collection of subspaces spanning the entire space. Each statement utilizes different information invariant to Lorentz space. These statements are crafted in Lorentz space with the intention of interpreting them in other geometric spaces.In the third chapter, a dictionary between the objects and tools of Lorentz $(n+1)$-space and those in hyperbolic $n$-space is outlined so that the rigidity results in chapter $2$ may be interpreted within the hyperbolic setting. Much of this information is standard and found within any given hyperbolic geometry text; some observations about the correspondence between linear independent vectors in Lorentz space and objects in hyperbolic space are novel. The Lorentz ratio also yields an invariant in hyperbolic space we call the hyperbolic ratio. The main rigidity result of objects in hyperbolic space is stated at the end of the chapter.In the final chapter, we turn our attention to the correspondence between objects in Lorentz space, and points and spheres in the $(n-1)$-sphere. There is an immediate correspondence that arises from the fact that the $(n-1)$-sphere can be taken as the ideal boundary of hyperbolic $n$-space. More specifically, circles and points in the Riemann sphere enjoy a geometry not dissimilar to points, lines, and planes in Euclidean space, so a notion of independence may be established within this geometry. This terminology is used to give a rigidity result for configurations of intermingled points and circles with independent collections of circles in the $2$-sphere. Whereas inversive distance is a common conformal invariant used between pairs of circles, an inversive ratio between a point and two circles is defined so that these intermingled configurations may be considered. This chapter ends with results on the rigidity of inversive distance circle packings that use independence as a tool.

ISBN: 9798678122179Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Circle Packing

Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry.
LDR:04879nmm a2200385 4500 001 2280406
005 20210827095942.5
008 220723s2020 ||||||||||||||||| ||eng d
020 $a 9798678122179
035 $a (MiAaPQ)AAI28002480
035 $a AAI28002480
040 $a MiAaPQ $c MiAaPQ
100 1 $a Graham, Opal. $0 (orcid)0000-0001-5259-5294 $3 3558922
245 1 0 $a Rigidity of Point and Sphere Configurations: An Examination of Rigidity in Lorentz, Hyperbolic, Euclidean, and Spherical Geometry.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2020
300 $a 94 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-05, Section: B.
500 $a Advisor: Bowers, Philip L.
502 $a Thesis (Ph.D.)--The Florida State University, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a This dissertation examines the study of rigidity of collections of objects in various geometric spaces, and the correspondences shared between geometries. In particular, we take a look at vectors and lines in Lorentz $(n+1)$-space, points, ideal points and hyperplanes in hyperbolic $n$-space, and circles and points in the Riemann sphere. One main objective is to explore how much information invariant to a given space is sufficient for a collection to be unique up to the transformations of the space. The answer to this question changes with the qualities a collection of objects possesses. To this end, this dissertation focuses on the role independence of objects plays in uniqueness. As another primary focus, a new invariant is introduced in each geometric setting to provide a means with which to study the rigidity of intermingled collections of objects that are infinitely far away from one another.The first chapter gives a history of circle, sphere, and point configurations, and the correspondences between configurations of objects in hyperbolic space and Lorentz space. All theorems stated in this chapter are well-established results and provide both motivation for studying conditions for rigidity and for introducing an invariant that allows for intermingled collections of points and spheres.In the second chapter, the main result is established entirely within the context of Lorentz space. An invariant of Lorentz transformations called the Lorentz ratio is defined that allows one to work with intermingled collections of vectors and light-like lines in Lorentz space. Three main statements are made about the rigidity of intermingled collections of vectors and light-like lines, each relying on a linearly independent collection of subspaces spanning the entire space. Each statement utilizes different information invariant to Lorentz space. These statements are crafted in Lorentz space with the intention of interpreting them in other geometric spaces.In the third chapter, a dictionary between the objects and tools of Lorentz $(n+1)$-space and those in hyperbolic $n$-space is outlined so that the rigidity results in chapter $2$ may be interpreted within the hyperbolic setting. Much of this information is standard and found within any given hyperbolic geometry text; some observations about the correspondence between linear independent vectors in Lorentz space and objects in hyperbolic space are novel. The Lorentz ratio also yields an invariant in hyperbolic space we call the hyperbolic ratio. The main rigidity result of objects in hyperbolic space is stated at the end of the chapter.In the final chapter, we turn our attention to the correspondence between objects in Lorentz space, and points and spheres in the $(n-1)$-sphere. There is an immediate correspondence that arises from the fact that the $(n-1)$-sphere can be taken as the ideal boundary of hyperbolic $n$-space. More specifically, circles and points in the Riemann sphere enjoy a geometry not dissimilar to points, lines, and planes in Euclidean space, so a notion of independence may be established within this geometry. This terminology is used to give a rigidity result for configurations of intermingled points and circles with independent collections of circles in the $2$-sphere. Whereas inversive distance is a common conformal invariant used between pairs of circles, an inversive ratio between a point and two circles is defined so that these intermingled configurations may be considered. This chapter ends with results on the rigidity of inversive distance circle packings that use independence as a tool.
590 $a School code: 0071.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Circle Packing
653 $a Discrete Conformal Geometry
653 $a Hyperbolic Geometry
653 $a Inversive Geometry
653 $a Lorentz Space
653 $a Rigidity
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a The Florida State University. $b Mathematics. $3 2095897
773 0 $t Dissertations Abstracts International $g 82-05B.
790 $a 0071
791 $a Ph.D.
792 $a 2020
793 $a English
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28002480