Date of Award
Doctor of Philosophy (PhD)
Euler's elastica, Graph embedding, Membrane shape, Pinned rigid graph, Rigid graph
Condensed Matter Physics | Physical Sciences and Mathematics | Physics
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 (
Kim, Kyung Eun, "Geometry of discrete and continuous bounded surfaces" (2022). Dissertations - ALL. 1656.