| dc.identifier.uri | http://hdl.handle.net/11401/77322 | |
| dc.description.sponsorship | This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. | en_US |
| dc.format | Monograph | |
| dc.format.medium | Electronic Resource | en_US |
| dc.language.iso | en_US | |
| dc.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dc.type | Dissertation | |
| dcterms.abstract | This thesis introduces the Ricci Flow and Its applications. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. General Ricci flow is defined on arbitrary dimensional Riemannian manifolds. Surface (2-manifold) Ricci flow has unique characteristics, which are crucial for developing discrete theories and designing computational algorithms. The unified theoretic framework for discrete Surface Ricci Flow is innovated, including all the common schemes: Tangential Circle Packing, Thurston's Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, we also introduce novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the unified framework. It gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Ricci flow has a lot of applications. Some are introduced in this thesis. First, Combine the Ricci flow and koebe's iteration for computing the canonical mapping for uniformizaiton of open surfaces, and give the theoretical proof of convergence. Second, a novel shape signature based on surface Ricci flow and optimal mass transportation is introduced for the purpose of surface comparison. Third, consider Ricci flow as conformal visualization technique and applied to immersive systems such as the CAVE. We can establishes a conformal mapping between the full 360 degree field of view and the display geometry of a given visualization system. The major challenges of visualizing the abstract Ricci curvature are to represent the intrinsic Riemannian metric of a surface by extrinsic embedding in the three dimensional Euclidean space and demonstrate the deformation process which preserves the conformal structure. A series of rigorous and practical algorithms are introduced to tackle the problem. | |
| dcterms.available | 2017-09-20T16:52:30Z | |
| dcterms.contributor | Gao, Jie | en_US |
| dcterms.contributor | Gu, Xianfeng | en_US |
| dcterms.contributor | Lu, Long | en_US |
| dcterms.contributor | Luo, Feng | en_US |
| dcterms.contributor | Gu, Xianfeng. | en_US |
| dcterms.creator | Zhang, Min | |
| dcterms.dateAccepted | 2017-09-20T16:52:30Z | |
| dcterms.dateSubmitted | 2017-09-20T16:52:30Z | |
| dcterms.description | Department of Computer Science. | en_US |
| dcterms.extent | 150 pg. | en_US |
| dcterms.format | Application/PDF | en_US |
| dcterms.format | Monograph | |
| dcterms.identifier | http://hdl.handle.net/11401/77322 | |
| dcterms.issued | 2014-12-01 | |
| dcterms.language | en_US | |
| dcterms.provenance | Made available in DSpace on 2017-09-20T16:52:30Z (GMT). No. of bitstreams: 1
Zhang_grad.sunysb_0771E_12003.pdf: 34370742 bytes, checksum: 126dbc5d12926a44bc1edaecce38cef7 (MD5)
Previous issue date: 1 | en |
| dcterms.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dcterms.subject | Computer science | |
| dcterms.subject | Conformal Mapping, Hessian Matrix, Metric, Poincar\'e conjecture, Ricci Flow, Uniformazation | |
| dcterms.title | Ricci Flow and Its Applications | |
| dcterms.type | Dissertation | |
