Skeleton Structures and Origami Design

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\u27s 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\u27\u27 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\u27\u27 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\u27s 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

Skeletal representations of orthogonal shapes

Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods

Incremental Convex Planarity Testing

AbstractAn important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to on-line insertions of vertices and edges. We present a data structure for the on-line incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worst-case time, insertion of vertices takes O(log n) worst-case time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n)

Generalized offsetting of planar structures using skeletons

We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure