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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Computer Science

Year Degree Awarded

2015

Month Degree Awarded

September

First Advisor

Ileana Streinu

Second Advisor

Andrew McGregor

Third Advisor

Gerome Miklau

Fourth Advisor

Tom Braden

Subject Categories

Geometry and Topology | Graphics and Human Computer Interfaces | Theory and Algorithms

Abstract

In this dissertation we study problems related to polygonal skeleton structures that have applications to computational origami. The two main structures studied are the straight skeleton of a simple polygon (and its generalizations to planar straight line graphs) and the universal molecule of a Lang polygon. This work builds on results completed jointly with my advisor Ileana Streinu.

Skeleton structures are used in many computational geometry algorithms. Examples include the medial axis, which has applications including shape analysis, optical character recognition, and surface reconstruction; and the Voronoi diagram, which has a wide array of applications including geographic information systems (GIS), point location data structures, motion planning, etc.

The straight skeleton, studied in this work, has applications in origami design, polygon interpolation, biomedical imaging, and terrain modeling, to name just a few. Though the straight skeleton has been well studied in the computational geometry literature for over 20 years, there still exists a significant gap between the fastest algorithms for constructing it and the known lower bounds.

One contribution of this thesis is an efficient algorithm for computing the straight skeleton of a polygon, polygon with holes, or a planar straight-line graph given a secondary structure called the induced motorcycle graph.

The universal molecule is a generalization of the straight skeleton to certain convex polygons that have a particular relationship to a metric tree. It is used in Robert Lang's seminal TreeMaker method for origami design. Informally, the universal molecule is a subdivision of a polygon (or polygonal sheet of paper) that allows the polygon to be ``folded'' into a particular 3D shape with certain tree-like properties. One open problem is whether the universal molecule can be rigidly folded: given the initial flat state and a particular desired final ``folded'' state, is there a continuous motion between the two states that maintains the faces of the subdivision as rigid panels? A partial characterization is known: for a certain measure zero class of universal molecules there always exists such a folding motion. Another open problem is to remove the restriction of the universal molecule to convex polygons. This is of practical importance since the TreeMaker method sometimes fails to produce an output on valid input due the convexity restriction and extending the universal molecule to non-convex polygons would allow TreeMaker to work on all valid inputs. One further interesting problem is the development of faster algorithms for computing the universal molecule. In this thesis we make the following contributions to the study of the universal molecule. We first characterize the tree-like family of surfaces that are foldable from universal molecules. In order to do this we define a new family of surfaces we call Lang surfaces and prove that a restricted class of these surfaces are equivalent to the universal molecules. Next, we develop and compare efficient implementations for computing the universal molecule. Then, by investigating properties of broader classes of Lang surfaces, we arrive at a generalization of the universal molecule from convex polygons in the plane to non-convex polygons in arbitrary flat surfaces. This is of both practical and theoretical interest. The practical interest is that this work removes the case from Lang's TreeMaker method that causes TreeMaker to fail to produce output in the presence of non-convex polygons. The theoretical interest comes from the fact that our generalization encompasses more than just those surfaces that can be cut out of a sheet of paper, and pertains to polygons that cannot be lied flat in the plane without self-intersections. Finally, we identify a large class of universal molecules that are not foldable by rigid folding motions. This makes progress towards a complete characterization of the foldability of the universal molecule.

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