Browsing by Subject "differential geometry"
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(2024)This thesis studies methods for finding crease patterns for surfaces of revolution with different Gaussian curvatures using variations of the Miura-ori origami pattern. Gaussian curvature is an intrinsic property of a surface in that it depends only on the inner properties of the surface in question. Usually determining the Gaussian curvature of a surface can be difficult, but for surfaces of revolution it can be calculated easily. Examples of surfaces of revolution with different Gaussian curvatures include cylinders, spheres, catenoids, pseudospheres and tori, which are the surfaces of interest in the work. Miura-ori is a family of flat-foldable origami patterns which consist of a quadrilateral mesh. The regular pattern is a two-way periodic tessellation which is determined by the parameters around a single vertex and it has a straight profile in all of its the semi-folded forms. By relaxing the pattern to a one-way periodic tessellation we get a more diverse set of patterns called the semi-generalized Miura-ori (SGMO) which are determined by the parameters of single column of vertices. By varying the angles of the creases related to these vertices we are also able to approximate curved profiles. Patterns for full surfaces of revolution can then be found by folding a thin strip of paper to an SGMO configuration that follows a wanted profile, after which the strip is repeated enough times horizontally to be able to join the ends of the paper to form a full revolution. Three algorithms for finding a crease pattern that follows a wanted profile curve are discussed in the work. This includes a simple algorithm by Robert J. Lang in addition to two algorithms developed by the author called the Equilateral triangles method and the Every second major fold follows the curve method. All three algorithms are explored both geometrically and by their pen-and-paper implementations which are described in detail so that the reader can utilize them without making any computations. Later, the three algorithms are tested on a set of profile curves for the surfaces of interest. Examples of full surfaces folded in real life are also given and the crease patterns for the models are included. The results showcase that each algorithm is suitable for finding patterns for our test set of surfaces and they usually have visually distinct appearances. The scale and proportions of the approximation matter greatly in terms of looks and feasibility of the pattern with all algorithms.
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(2021)HMC is a computational method build to efficiently sample from a high dimensional distribution. Sampling from a distribution is typically a statistical problem and hence a lot of works concerning Hamiltonian Monte Carlo are written in the mathematical language of probability theory, which perhaps is not ideally suited for HMC, since HMC is at its core differential geometry. The purpose of this text is to present the differential geometric tool's needed in HMC and then methodically build the algorithm itself. Since there is a great introductory book to smooth manifolds by Lee and not wanting to completely copy Lee's work from his book, some basic knowledge of differential geometry is left for the reader. Similarly, the author being more comfortable with notions of differential geometry, and to cut down the length of this text, most theorems connected to measure and probability theory are omitted from this work. The first chapter is an introductory chapter that goes through the bare minimum of measure theory needed to motivate Hamiltonian Monte Carlo. Bulk of this text is in the second and third chapter. The second chapter presents the concepts of differential geometry needed to understand the abstract build of Hamiltonian Monte Carlo. Those familiar with differential geometry can possibly skip the second chapter, even though it might be worth while to at least flip through it to fill in on the notations used in this text. The third chapter is the core of this text. There the algorithm is methodically built using the groundwork laid in previous chapters. The most important part and the theoretical heart of the algorithm is presented here in the sections discussing the lift of the target measure. The fourth chapter provides brief practical insight to implementing HMC and also discusses quickly how HMC is currently being improved.
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(2023)The thesis consists of presenting and analysing the original proof for the Embedding Theorem that Hassler Whitney gave in his 1936 article Differentiable Manifolds. The embedding theorem states that given an m-dimensional Cr-differentiable (r ≥ 1) manifold M, it is possible to embed it in Euclidean space Rn, if n ≥ 2m + 1. Embedding is defined as a mapping f : M → Rn which is Cr-smooth, bijective immersion that is homeomorphism to its image f[M]. Whitney’s proof rests on few important novel concepts and a series of lemmas in relation to them. These concepts include the concept of the k-extent of a set, a sort of a k-dimensional measure in an n-dimensional space; the concept of Cr-function g : M → N approximating (f, M, r, η), where f is a Cr-function f : M → N, η an error function; and the concept of (f, r, η)-properties defined for such g. Outstanding lemmas of general nature are Lemma 7: If f : M → N is a Cr-map and A ⊂ M is of finite (zero) k-extent, then f[A] is of finite (zero) k-extent. Lemma 8: For open sets R and R′ of Rm and Rh, if {Tα} is a h-parameter family of C1-maps of R ⊂ Rm into Rn, and A ⊂ R and B ⊂ Rn closed subsets, such that A is of finite k-extent and B of zero (h − k)-extent, then for some α ∈ R′, Tα[A] does not intersect B. Lemma 9: If f : M → N is a Cr-map, η positive continuous function in M, Ω1, Ω2, . . . are (f, r, η)- properties, then there is a Cr-map F : M → N which approximates (f, M, r, η) and has properties Ω1, Ω2, . . . . Lemmas 11 and 12 then show that bijectivity and immersion property are the logical sum of countable number of (f, r, η)-properties. These facts are used in finding an embedding F : M → Rn by perturbing a given smooth function f : M → Rn. Detailed treatment of all proofs is provided. Adjustments to the proofs are made where deemed necessary; auxiliary assumptions are made where they seem to be required. Clarifications and proofs are given to facts noted but not proven in the article
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