Date of Award
8-26-2022
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Physics
Advisor(s)
Christian Santangelo
Second Advisor
Wanliang Shan
Keywords
Euler's elastica, Graph embedding, Membrane shape, Pinned rigid graph, Rigid graph
Subject Categories
Condensed Matter Physics | Physical Sciences and Mathematics | Physics
Abstract
We work on reconstructing discrete and continuous surfaces with boundaries using length constraints. First, for a bounded discrete surface, we discuss the rigidity and number of embeddings in three-dimensional space, modulo rigid transformations, for given real edge lengths. Our work mainly considers the maximal number of embeddings of rigid graphs in three-dimensional space for specific geometries (annulus, strip). We modify a commonly used semi-algebraic, geometrical formulation using Bézout's theorem, from Euclidean distances corresponding to edge lengths. We suggest a simple way to construct a rigid graph having a finite upper bound. We also implement a generalization of counting embeddings for graphs by segmenting multiple rigid graphs in d-dimensional space. Our computational methodology uses vector and matrix operations and can work best with a relatively small number of points (
Access
Open Access
Recommended Citation
Kim, Kyung Eun, "Geometry of discrete and continuous bounded surfaces" (2022). Dissertations - ALL. 1656.
https://surface.syr.edu/etd/1656
