University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses November 2015 Skeleton Structures and Origami Design John C. Bowers University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Geometry and Topology Commons, Graphics and Human Computer Interfaces Commons, and the Theory and Algorithms Commons Recommended Citation Bowers, John C., "Skeleton Structures and Origami Design" (2015). Doctoral Dissertations. 477. https://doi.org/10.7275/7463101.0 https://scholarworks.umass.edu/dissertations_2/477 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. SKELETON STRUCTURES AND ORIGAMI DESIGN A Dissertation Presented by JOHN CHRISTOPHER BOWERS Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2015 School of Computer Science c Copyright by John Christopher Bowers 2015 All Rights Reserved SKELETON STRUCTURES AND ORIGAMI DESIGN A Dissertation Presented by JOHN CHRISTOPHER BOWERS Approved as to style and content by: Ileana Streinu, Chair Andrew McGregor, Member Gerome Miklau, Member Tom Braden, Member Lori Clarke, Chair School of Computer Science DEDICATION For Katherine and Pippin and Scout. ACKNOWLEDGMENTS I would like to express my heartfelt thanks and appreciation for the many people who helped me in completing this thesis. This thesis would not have been possible without the advice and support of my advisor, Ileana Streinu. Ileana's adept guidance was crucial to my maturing as a researcher, a scholar, and a teacher. From her I learned perseverance (especially when facing reviewers), confidence and independence, and above all how to be a researcher. When I needed it, she pushed me (especially to revise, revise, revise!), and when I needed to step up on my own, she gave me free rein to do so. Most importantly, she introduced and fostered in me the joy of computational geometry. I remember coming away from one of the first lectures in Ileana's class thinking, this is it; this is what I want to study. I am very grateful for her mentorship and greatly value my time working with and learning from her. I also want to thank my committee, Andrew McGregor, Gerome Miklau, and Tom Braden, for their support and guidance. Additionally, I thank Andrew for many helpful conversations over the course of my graduate career. I thank Tom for helping me to improve the writing of this document. I next want to thank Rui Wang, who helped me get my start as a researcher, took me to my first academic conferences, and has been an inspiration and a support throughout my graduate career. Rui's energy and enthusiasm is infectious and he is a model for the type of researcher and advisor I want to be. I hope that my interactions with future students will be as energetic and inspiring as Rui's were with me. Especially in the last year, I have benefitted greatly from the professional advice and sound council of many that traveled this path before me. I thank Andrew Berke, v Henry Field, Jackie Field, Megan Olson, Audrey St. John, and Jerod Weinman for encouragement at the last mile, for proofreading, and especially for advice that gave me the courage to complete this thesis and start my career as a professor. I want to thank the many Theory and LinKaGe Lab students who were in the trenches with me: Ashraf Alam, Marco Carmosino, Michael Crouch, Naomi Fox, Cibele Freire, Filip Jagodzinski, Brandon McPhail, Daniel Stubbs, Patrick Taylor, David Tench, Hoa Vu, and Sofya Vorotnikova. Through conversation, debate, white- board mini-lectures, practice talks, and plenty of coffee, I found encouragement, in- spiration, and above all kept my sanity. A special thanks as well to Leeanne Leclerc, who deftly shepherded me (and many before me) through the many hoops and re- quirements of the graduate school. A large part of this work was supported by the National Science Foundation Graduate Research Fellowship Program. I first started down the path of a computer scientist when a friend of my parents, Steve Payne, taught me to program in sixth grade. Since then Steve has become both a mentor and a close friend. I am very much in debt to Steve for the years of support, guidance, and prayers. I remember at one point during high school a warning he gave me that I (mostly) took to heart. He warned me it sometimes becomes necessary to focus on one direction and not to simply become a Jack of All Trades and Master of None. This thesis is my heeding of that advice. I also thank those many friends who have given me love and support throughout this thesis process; for the friendship and support of Craig Nicolson{he has been a invaluable fellow traveller in The Way and additionally helped me to grow as a presenter and speaker; for Ian Callahan, whose courage and steadfastness inspires me daily; for Mike Foster and Richard Vachet, who model the sort of professor I want to be; for Steve Oloo, who has shared the PhD journey with me; for Ben Greene, who adventured into forest, over cliff, and up ice flows, when I needed to escape the LED vi glare of my computer screen; and for Nate Daman, who has been a parenting role model, friend, and council. Above all I gratefully thank my wife, Katherine Bowers, and my family. This thesis is a part of my and Katherine's shared vocation and I thank her for her patience and support throughout the process of completing it. She is my best friend, and is a wonderful mother to our little ones. This thesis is dedicated to her and to our two children, Pippin and Scout, who are my daily joy and inspiration. I thank my parents, Phil and Kris Bowers, for the innumerable ways they have supported me and nurtured me{to Dad, especially for the many mathematical conversations and coffee supply; and to Mom, especially for the prayers and encouragement and teaching advice. I thank my siblings, Maddy and Thomas Bowers, for relaxing and recreating with me. I thank my parents-in-law, Eric and Lu Grimm, and my sisters-in-law, Kellie Bowers and Beca Grimm, for the years of encouragement, and countless ways they have made me feel supported and loved. vii ABSTRACT SKELETON STRUCTURES AND ORIGAMI DESIGN SEPTEMBER 2015 JOHN CHRISTOPHER BOWERS B.Sc., THE FLORIDA STATE UNIVERSITY M.Sc., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ileana Streinu 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. Exam- ples 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 viii 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 con- vex 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 univer- sal 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 uni- versal 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 out- put 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.