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Grid Vertex-Unfolding Orthogonal Polyhedra
[chapter]

2006
*
Lecture Notes in Computer Science
*

We show that any

doi:10.1007/11672142_21
fatcat:ljnjvqnidrhhzck3m27b7npmqm
*orthogonal**polyhedra*of genus zero has a*grid**vertex*-*unfolding*. (There are*orthogonal**polyhedra*that cannot be*vertex*-*unfolded*, so some type of "*gridding*" of the faces is necessary.) ... A*grid**unfolding*allows additional cuts along*grid*edges induced by coordinate planes passing through every*vertex*. ... For*orthogonal**polyhedra*, a*grid**unfolding*is a natural median between edge-*unfoldings*and unrestricted*unfoldings*. ...##
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Grid Vertex-Unfolding Orthogonal Polyhedra
[article]

2006
*
arXiv
*
pre-print

(There are

arXiv:cs/0509054v2
fatcat:lb2267luovblja6rtsakiqh75y
*orthogonal**polyhedra*that cannot be*vertex*-*unfolded*, so some type of "*gridding*" of the faces is necessary.) ... A*vertex*-*unfolding*permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any*orthogonal*polyhedron of genus zero has a*grid**vertex*-*unfolding*. ... For*orthogonal**polyhedra*, a*grid**unfolding*is a natural median between edge-*unfoldings*and unrestricted*unfoldings*. ...##
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Grid Vertex-Unfolding Orthogonal Polyhedra

2007
*
Discrete & Computational Geometry
*

We show that any

doi:10.1007/s00454-007-9043-9
fatcat:qiroad7z3ffaxp355gitbsrgqy
*orthogonal**polyhedra*of genus zero has a*grid**vertex*-*unfolding*. (There are*orthogonal**polyhedra*that cannot be*vertex*-*unfolded*, so some type of "*gridding*" of the faces is necessary.) ... A*grid**unfolding*allows additional cuts along*grid*edges induced by coordinate planes passing through every*vertex*. ... For*orthogonal**polyhedra*, a*grid**unfolding*is a natural median between edge-*unfoldings*and unrestricted*unfoldings*. ...##
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Grid Vertex-Unfolding Orthogonal Polyhedra
[chapter]

*
Twentieth Anniversary Volume:
*

We show that any

doi:10.1007/978-0-387-87363-3_13
fatcat:3ztkxklt4nhxrcknsh2q5etb3i
*orthogonal**polyhedra*of genus zero has a*grid**vertex*-*unfolding*. (There are*orthogonal**polyhedra*that cannot be*vertex*-*unfolded*, so some type of "*gridding*" of the faces is necessary.) ... A*grid**unfolding*allows additional cuts along*grid*edges induced by coordinate planes passing through every*vertex*. ... For*orthogonal**polyhedra*, a*grid**unfolding*is a natural median between edge-*unfoldings*and unrestricted*unfoldings*. ...##
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Vertex Unfoldings of Orthogonal Polyhedra: Positive, Negative, and Inconclusive Results

2018
*
Canadian Conference on Computational Geometry
*

The (positive) first result is a simple proof that all genus-0 and genus-1

dblp:conf/cccg/GarciaGRW18
fatcat:rbvni4mpnnd27gmeumpogpk6ua
*orthogonal**polyhedra*have*grid**vertex**unfoldings*. ... We obtain results for three questions regarding*vertex**unfoldings*of*orthogonal**polyhedra*. ... Acknowledgments We would like to thank Erik Demaine for discussion of known results related to*vertex**unfoldings*, and anonymous reviewers whose suggestions improved the readability and correctness of the ...##
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Unfolding Orthogonal Terrains
[article]

2007
*
arXiv
*
pre-print

It is shown that every

arXiv:0707.0610v4
fatcat:4ljvfcklb5elfdbihtn4yrms2u
*orthogonal*terrain, i.e., an*orthogonal*(right-angled) polyhedron based on a rectangle that meets every vertical line in a segment, has a*grid**unfolding*: its surface may be*unfolded*... to a single non-overlapping piece by cutting along*grid*edges defined by coordinate planes through every*vertex*. ... it may be that all*orthogonal**polyhedra*may be*grid*-*unfoldable*. ...##
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Improved Algorithms for Grid-Unfolding Orthogonal Polyhedra

2017
*
International journal of computational geometry and applications
*

For 1-layer

doi:10.1142/s0218195917600032
fatcat:7obx26no3rbi5gwmk4rirgx5ta
*orthogonal**polyhedra*of genus g, we show a*grid*-*unfolding*algorithm using only 2(g − 1) additional cuts, affirmatively answering an open problem raised in a recent literature. ... Even for*orthogonal**polyhedra*, it is known that edge-*unfolding*, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful*unfolding*in general. ... Can all*orthogonal**polyhedra*be*grid*-*unfolded*with sublinear refinement? ...##
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Unfolding Level 1 Menger Polycubes of Arbitrary Size With Help of Outer Faces
[chapter]

2019
*
Lecture Notes in Computer Science
*

It is worth noticing that this

doi:10.1007/978-3-030-14085-4_36
fatcat:ojow7z2kzve4ti7ilabk3ejyxy
*grid*-*unfolding*algorithm is deterministic and without refinement. ... In this article, we suggest a*grid*-*unfolding*of level 1 Menger polycubes of arbitrary size with L holes along the x-axis, M the y-axis and N the z-axis. ... In [10] , edge-*unfolding*of several classes of one-layered*orthogonal**polyhedra*(one layer of voxels) with cubic holes are considered and it was shown that several special classes of them admit*grid**unfoldings*...##
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Epsilon-Unfolding Orthogonal Polyhedra

2007
*
Graphs and Combinatorics
*

Here we prove, via an algorithm, that every

doi:10.1007/s00373-007-0701-8
fatcat:4yisqxefrjdrvnxpmrfmex4k6u
*orthogonal*polyhedron (one whose faces meet at right angles) of genus zero may be*unfolded*. ... It is a long unsolved problem to determine whether every polyhedron may be*unfolded*. ... This concept has been used to achieve*grid**vertex**unfoldings*of orthostacks [3] , and later,*grid**vertex**unfoldings*of all genus-zero*orthogonal**polyhedra*[8] . ...##
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Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm
[article]

2011
*
arXiv
*
pre-print

We show that every

arXiv:1112.4791v1
fatcat:w4n7knv5y5ak3evvvrbithmoza
*orthogonal*polyhedron homeomorphic to a sphere can be*unfolded*without overlap while using only polynomially many (*orthogonal*) cuts. ... More precisely, given an*orthogonal*polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the*grid*of coordinate planes passing through the vertices, together with Theta ... However, our preliminary investigations embolden us to conjecture that a constant refinement of the*vertex**grid*suffices to*grid*-*unfold*all*orthogonal**polyhedra*. ...##
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Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

2012
*
Graphs and Combinatorics
*

We show that every

doi:10.1007/s00373-012-1257-9
fatcat:pae5s6hd7rhcdgf2mhq44apipa
*orthogonal*polyhedron homeomorphic to a sphere can be*unfolded*without overlap while using only polynomially many (*orthogonal*) cuts. ... More precisely, given an*orthogonal*polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the*grid*of coordinate planes passing through the vertices, together with Θ(n 2 ... However, our preliminary investigations embolden us to conjecture that a constant refinement of the*vertex**grid*suffices to*grid*-*unfold*all*orthogonal**polyhedra*. ...##
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Unfolding Manhattan Towers

2008
*
Computational geometry
*

The algorithm cuts along edges of a 4 × 5 × 1 refinement of the

doi:10.1016/j.comgeo.2007.07.003
fatcat:6mgpp5ojjrgqhf3qb2yhczhdtu
*vertex**grid*. ... We provide an algorithm for*unfolding*the surface of any*orthogonal*polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar*orthogonal*polygon. ... the suggestion was made in [4] to seek k 1 × k 2 × k 3 -refined*grid**unfoldings*, where every face of the*vertex**grid*is further refined into a*grid*of edges. ...##
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Unfolding Manhattan Towers
[article]

2007
*
arXiv
*
pre-print

The algorithm cuts along edges of a 4x5x1 refinement of the

arXiv:0705.1541v1
fatcat:snb4x7fbxvhuxhee6efizihyqu
*vertex**grid*. ... We provide an algorithm for*unfolding*the surface of any*orthogonal*polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar*orthogonal*polygon. ... the suggestion was made in [DO04] to seek k 1 × k 2 × k 3 refined*grid**unfoldings*, where every face of the*vertex**grid*is further refined into a*grid*of edges. ...##
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Epsilon-Unfolding Orthogonal Polyhedra
[article]

2006
*
arXiv
*
pre-print

Here we prove, via an algorithm, that every

arXiv:cs/0602095v2
fatcat:gfpo4rntfjhstig5hvgcnnei4e
*orthogonal*polyhedron (one whose faces meet at right angles) of genus zero may be*unfolded*. ... It is a long unsolved problem to determine whether every polyhedron may be*unfolded*. ... This concept has been used to achieve*grid**vertex**unfoldings*of orthostacks [DIL04] , and later,*grid**vertex**unfoldings*of all genus-zero*orthogonal**polyhedra*[DFO06] . ...##
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Grid Vertex-Unfolding Orthostacks
[chapter]

2005
*
Lecture Notes in Computer Science
*

It was conjectured that orthostacks could be

doi:10.1007/11589440_8
fatcat:riyfavfrnbh7jhvtdgy5p2ayqi
*unfolded*using cuts that lie in a plane*orthogonal*to a coordinate axis and containing a*vertex*of the orthostack. ... We prove the existence of a*vertex*-*unfolding*using only such cuts. ... We make partial progress on this problem by show- ing that every orthostack can be*grid**vertex*-*unfolded*, i.e., cut along some of the*grid*lines and*unfolded*into a*vertex*-connected planar piece without ...
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