Browsing by Subject "Gaussian curvature"

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.