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

Computer representation of graphical information with applications

PhD ThesisThe research work contained in this thesls lies mainly in the field of computer graphics. The initial chapters are concerned with methods of representing three dimensional solids in two dimensions. Chapter 2 describes a method by which points in three dimensions can be projected onto a two dimensional plane of This is an essential requirement in the projection. This is an essential requirement in the representation of three dimensional solids. Chapter 3 describes a method by which convex polyhedra can be represented by computer. Both the hidden polyhedra and visible face of the polyhedron can be represented by computer. Having tackled this problem, the more difficult problem of representing the non convex polyhedron has been attempted and the results of this work are presented in Chapter 4. Line drawings of the various polyhedra, produced on a graph plotter, are given as examples at the end of Chapters 2, 3 and 4. The problem of how to connect a given line drawing such that the distance travelled by the pen of some computer display is kept to a minimum is discussed in Chapter 5 and various definitions of the concepts involved are given. Theory associated with this 'Pen-Up Problem' has been developed and is explained in detail in the early part of Chapter 6. A method of obtaining an optimal solution to the problem is presented in the latter part of this chapter in addition to various enumerative schemes which have been developed to obtain good feasible solutions to the pen up problems under various conditions Extensive C.P.U. timing experiments have been carried out in Chapter 7 on the various enumerative schemes in Chapter 6 and it has introduced been possible to reach conclusions on the applicability of the various methods. Several topics of interest which have arisen during the main research work are presented as appendices. The programs which have been coded during the period of research are also inc1udeu as appendices

Efficient view point selection for silhouettes of convex polyhedra

AbstractSilhouettes of polyhedra are an important primitive in application areas such as machine vision and computer graphics. In this paper, we study how to select view points of convex polyhedra such that the silhouette satisfies certain properties. Specifically, we give algorithms to find all projections of a convex polyhedron such that a given set of edges, faces and/or vertices appear on the silhouette.We present an algorithm to solve this problem in O(k2) time for k edges. For orthogonal projections, we give an improved algorithm that is fully adaptive in the number l of connected components formed by the edges, and has a time complexity of O(klogk+kl). We then generalize this algorithm to edges and/or faces appearing on the silhouette

Extracting curve-skeletons from digital shapes using occluding contours

Curve-skeletons are compact and semantically relevant shape descriptors, able to summarize both topology and pose of a wide range of digital objects. Most of the state-of-the-art algorithms for their computation rely on the type of geometric primitives used and sampling frequency. In this paper we introduce a formally sound and intuitive definition of curve-skeleton, then we propose a novel method for skeleton extraction that rely on the visual appearance of the shapes. To achieve this result we inspect the properties of occluding contours, showing how information about the symmetry axes of a 3D shape can be inferred by a small set of its planar projections. The proposed method is fast, insensitive to noise, capable of working with different shape representations, resolution insensitive and easy to implement

Perceiving ribs in single-view wireframe sketches of polyhedral shapes

As part of a strategy for creating 3D models of engineering objects from sketched input, we attempt to identify design features, geometrical structures within objects with a functional meaning. Our input is a 2D B-Rep derived from a single view sketch of a polyhedral shape. In this paper, we show how to use suitable cues to identify algorithmically two additive engineering design features, angular and linear ribs

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website