A theory of discrete parametrized surfaces in R^3
by Andrew O'Shea Sageman-Furnas
Date of Examination:2017-10-19
Date of issue:2018-07-27
Advisor:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Alexander Bobenko
Persistent Address:
http://dx.doi.org/10.53846/goediss-6977
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Abstract
English
In discrete differential geometry (DDG) one considers objects from discrete geometry from a differential geometric perspective. Rather than focusing on approximations of the smooth theory, with error vanishing in the continuum limit, DDG focuses on theories that exactly preserve geometric quantities and/or the structure of governing equations at every finite resolution. This thesis is concerned with the DDG of parametrized surfaces in three dimensional Euclidean space represented as so-called quad nets, immersed two dimensional complexes with quadrilateral faces. Our main focus is to obtain analogs of differential geometric notions for quad nets with immersed quadrilaterals that are nonplanar. This thesis is split into two parts. In Part I, we introduce the theory of edge-constraint nets, a general discrete surface theory in R^3 that unites the most prevalent versions of discrete analogs of surfaces in special parametrizations. Our theory encapsulates a large class of discrete so-called integrable geometries and, in particular, provides geometric insight into the algebraically constructed discrete analogs of one-parameter associated families of constant curvature surfaces, by introducing notions for both curvature and conformality. Conformal equivalence is introduced for edge-constraint nets using a discrete analog of spin transformations, which is then used to construct discrete Bonnet pairs, two immersed surfaces that are isometric and have the same mean curvature, but are not congruent. Edge-constraint nets are not restricted to integrable geometries, but lay the foundation for a general surface theory that lifts the restriction to special parametrizations. In Part II, we apply DDG principles to design with inherently discrete materials built from regular grids of inextensible rods, ranging from densely woven wire mesh to sparse elastic gridshells. We model these materials using smooth and discrete Chebyshev nets, a special type of surface parametrization that encodes rod inextensibility. Analytical properties of Chebyshev nets impose counterintuitive global constraints, but taking an applied perspective leads to computational algorithms for interactive design tools and an understanding of the global nature of designing with such materials, whose results suggest a rich design space.
Keywords: discrete differential geometry