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Creating Optimized CutOut Sheets for Paper Models from
"... Rapid advancement in technology has made virtual 3D models popular and increasingly affordable. However, 3D displays alone are usually insufficient for a complete understanding of the virtual object. A physical model of the object is often required, and this could be laborious and expensive to produ ..."
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Rapid advancement in technology has made virtual 3D models popular and increasingly affordable. However, 3D displays alone are usually insufficient for a complete understanding of the virtual object. A physical model of the object is often required, and this could be laborious and expensive to produce. To build prototypes
EdgeUnfolding Orthogonal Polyhedra is Strongly NPComplete
"... We prove that it is strongly NPcomplete to decide whether a given orthogonal polyhedron has a (nonoverlapping) edge unfolding. The result holds even when the polyhedron is topologically convex, i.e., is homeomorphic to a sphere, has faces that are homeomorphic to disks, and where every two faces sh ..."
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We prove that it is strongly NPcomplete to decide whether a given orthogonal polyhedron has a (nonoverlapping) edge unfolding. The result holds even when the polyhedron is topologically convex, i.e., is homeomorphic to a sphere, has faces that are homeomorphic to disks, and where every two faces share at most one edge. 1
Continuous Blooming of Convex Polyhedra
, 906
"... We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming. 1 ..."
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We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming. 1
Optimized Topological Surgery for Unfolding 3D Meshes
"... Constructing a 3D papercraft model from its unfolding has been fun for both children and adults since we can reproduce virtual 3D models in the real world. However, facilitating the papercraft construction process is still a challenging problem, especially when the shape of the input model is comple ..."
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Constructing a 3D papercraft model from its unfolding has been fun for both children and adults since we can reproduce virtual 3D models in the real world. However, facilitating the papercraft construction process is still a challenging problem, especially when the shape of the input model is complex in the sense that it has large variation in its surface curvature. This paper presents a new heuristic approach to unfolding 3D triangular meshes without any shape distortions, so that we can construct the 3D papercraft models through simple atomic operations for gluing boundary edges around the 2D unfoldings. Our approach is inspired by the concept of topological surgery, where the appearance of boundary edges of the unfolded closed surface can be encoded using a symbolic representation. To fully simplify the papercraft construction process, we developed a geneticbased algorithm for unfolding the 3D mesh into a single connected patch in general, while optimizing the usage of the paper sheet and balance in the shape of that patch. Several examples together with user studies are included to demonstrate that the proposed approach works well for a broad range of 3D triangular meshes. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems mesh unfolding, topological surgery, genetic algorithms, papercraft models 1.
The Unfolding Problem
, 2005
"... 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 ..."
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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 edgetoedge 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 paperfolding [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 selfintersecting, solely by breaking a subset of its edges and flattening the join angles of those which remain, then it is called edgeunfoldable 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 nonconvex surfaces which are cannot be edgeunfolded, but no robust solution yet exists for testing whether or not a given mesh will prove to be developable.
A Topologically Convex Vertexununfoldable Polyhedron
, 2011
"... We construct a polyhedron that is topologically convex (i.e., has the graph of a convex polyhedron) yet has no vertex unfolding: no matter how we cut along the edges and keep faces attached at vertices to form a connected (hinged) surface, the surface necessarily unfolds with overlap. ..."
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We construct a polyhedron that is topologically convex (i.e., has the graph of a convex polyhedron) yet has no vertex unfolding: no matter how we cut along the edges and keep faces attached at vertices to form a connected (hinged) surface, the surface necessarily unfolds with overlap.
DETC201312692 JOINING UNFOLDINGS OF 3D SURFACES
"... Origamibased design methods enable complex devices to be fabricated quickly in plane and then folded into their final 3D shapes. So far, these folded structures have been designed manually. This paper presents a geometric approach to automatic composition of folded surfaces, which will allow exist ..."
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Origamibased design methods enable complex devices to be fabricated quickly in plane and then folded into their final 3D shapes. So far, these folded structures have been designed manually. This paper presents a geometric approach to automatic composition of folded surfaces, which will allow existing designs to be combined and complex functionality to be produced with minimal human input. We show that given two surfaces in 3D and their 2D unfoldings, a surface consisting of the two originals joined along an arbitrary edge can always be achieved by connecting the two original unfoldings with some additional linking material, and we provide an algorithm to generate this composite unfolding. The algorithm is verified using various surfaces, as well as a walking and gripping robot design. 1
Noname manuscript No. (will be inserted by the editor) Continuous Blooming of Convex Polyhedra
"... Abstract We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming. 1 ..."
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Abstract We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming. 1