Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

In this paper, we study unfoldings of orthogonal polyhedra. More precisely, we define two special classes of orthogonal polyhedra, orthostacks and orthotubes, and show how to generate unfoldings by cutting faces, such that the resulting surfaces can be flattened into a single connected polygon.

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.

We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algorithm uses a new approach which deviates from the conventional "continuous Dijkstra" technique. Our algorithm has time complexity O(n²) and space complexity \Theta (n).

author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.

Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Θ(mn²), and the algorithm runs in O(mn² log² mlogn) time.

Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v ∈ S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R 2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex manifolds.

this paper, we study the problem of whether a polyhedron can be obtained from a net , i.e., a polygon and a set of creases, by folding along the creases. We consider two cases, depending on whether we are given the dihedral angle at each crease. If these dihedral angles are given the problem can be solved in polynomial time by the simple expedient of performing the folding. If the dihedral angles are not given the problem is NP-complete, at least for orthogonal polyhedra. We then turn to the actual folding process, and show an example of a net with rigid faces that can, in the sense above, be folded to form an orthogonal polyhedron, but only by allowing faces to intersect each other during the folding process.