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?

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 show that the decision question posed in the title can be answered with an algorithm of time and space complexity O(n 2 ), for a polygon of n vertices. We use a theorem of Aleksandrov that says that if the edges of the polygon can be matched in length so that the resulting complex is homeomorphic to a sphere, and such that the "complete angle" at each vertex is no more than 2, then the implied folding corresponds to a unique convex polytope. We check the Aleksandrov conditions via dynamic programming. The algorithm has been implemented and tested. 1 Introduction The polygon shown in Fig. 1a cannot fold edge-to-edge to a convex polytope despite being composed of six squares that one might think could fold to a cube. The familiar "cross" polygon in Fig. 1b can of course fold to a cube. The aim of this paper is to provide an algorithm that can decide whether a polygon can fold to a polytope. We take a polygon P to be a collection of vertices (v 0 ; v 1 ; :::; v n\Gamma1 ) in...

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.

We have studied the accuracy of statistical parameters of dose distributions in brachytherapy using actual clinical implants. These include the mean, minimum and maximum dose values and the variance of the dose distribution inside the PTV and on the surface of the PTV. These properties have been studied as a function of the number of uniformly distributed sampling points. These parameters, or the variants of these parameters, are used directly or indirectly in optimization procedures or for a description of the dose distribution. The accurate determinaton of these parameters depends on the sampling point distribution from which they have been obtained.

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints...

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The Unfolding Problem can be succinctly described as “How to peel an orange... no matter what shape the orange is. ” It is the question of how to ‘unwrap ’ a 3D polyhedron, breaking some of its edges or faces so that it can be unfolded into a flat net in the 2D plane. From there the flattened net of faces might be printed out, cut from paper or steel and folded to recreate the virtual model in the real world. In 1525 the artist Albrecht Dürer used the term ‘net ’ to describe a set of polygons linked together edge-to-edge to form the planar unfoldings of some of the platonic solids and their truncations. Dürer used these unfoldings to teach aspiring artists how to construct elemental forms, but today the applications for solutions to the unfolding problem lie in a broad range of fields, from industrial manufacturing and rapid prototyping to sculpture and aeronautics. In the textiles industry, work has already begun in computing digital representations of fabric and trying to flatten those representations to optimize seam and dart placement [MHC05]. There has been similar work in the fields of paper-folding [MS04] and origami [BM04] and even ship and sail manufacturing. Advances in robotics and folding automation [GBKK98] have brought with them a new need for faster, more robust unfolding methods. If a polyhedron can generate a net which is not self-intersecting, solely by breaking a subset of its edges and flattening the join angles of those which remain, then it is called edge-unfoldable or developable. At present, it is strongly believed–but not yet proven–that all convex surfaces are developable. In counterpoint, examples are easily found of non-convex surfaces which are cannot be edge-unfolded, but no robust solution yet exists for testing whether or not a given mesh will prove to be developable.