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Articles 91 - 106 of 106

Full-Text Articles in Geometry and Topology

On Reconfiguring Tree Linkages: Trees Can Lock, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides Sep 2000

On Reconfiguring Tree Linkages: Trees Can Lock, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides

Computer Science: Faculty Publications

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N-link tree can have 2Ω(N) equivalence classes of configurations.


Pushpush And Push-1 Are Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke Sep 2000

Pushpush And Push-1 Are Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that two pushing-blocks puzzles are intractable in 2D. One of our constructions improves an earlier result that established intractability in 3D [OS99] for a puzzle inspired by the game PushPush. The second construction answers a question we raised in [DDO00] for a variant we call Push-1. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a …


Computational Geometry Column 39, Joseph O'Rourke Aug 2000

Computational Geometry Column 39, Joseph O'Rourke

Computer Science: Faculty Publications

The resolution of a decades-old open problem is described: polygonal chains cannot lock in the plane.


Examples, Counterexamples, And Enumeration Results For Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke Jul 2000

Examples, Counterexamples, And Enumeration Results For Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke

Computer Science: Faculty Publications

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is …


Computational Geometry Column 38, Joseph O'Rourke Apr 2000

Computational Geometry Column 38, Joseph O'Rourke

Computer Science: Faculty Publications

Recent results on curve reconstruction are described.


Computational Geometry Column 37, Erik D. Demaine, Joseph O'Rourke Feb 2000

Computational Geometry Column 37, Erik D. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

Open problems from the 15th Annual ACM Symposium on Computational Geometry.


Pushpush Is Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke Jan 2000

Pushpush Is Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that a particular pushing-blocks puzzle is intractable in 2D, improving an earlier result that established intractability in 3D [OS99]. The puzzle, inspired by the game *PushPush*, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 2D by reduction from SAT.


Computational Geometry Column 36, Joseph O'Rourke Dec 1999

Computational Geometry Column 36, Joseph O'Rourke

Computer Science: Faculty Publications

Two results in "computational origami" are illustrated.


Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group Nov 1999

Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group

Computer Science: Faculty Publications

We prove that a particular pushing-blocks puzzle is intractable in 3D. The puzzle, inspired by the game PushPush, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 3D by reduction from SAT. The corresponding problem in 2D remains open.


Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko Jun 1999

Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko

Computer Science: Faculty Publications

Everett et al. [EHN96, EHN97] introduced several varieties of stabbing information for the lines determined by pairs of vertices of a simple polygon P, and established their relationships to vertex visibility and other combinatorial data. In the same spirit, we define the “zero-parity (ZP) stabbing information” to be a natural weakening of their “weak stabbing information,” retaining only the distinction among {zero, odd, even > 0} in the number of polygon edges stabbed. Whereas the weak stabbing information’s relation to visibility remains an open problem, we completely settle the analogous questions for zero parity information, with three results: (1) ZP information …


Locked And Unlocked Polygonal Chains In 3d, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Mark Overmars, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides Jan 1999

Locked And Unlocked Polygonal Chains In 3d, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Mark Overmars, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides

Computer Science: Faculty Publications

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a …


Computational Geometry Column 35, Joseph O'Rourke Jan 1999

Computational Geometry Column 35, Joseph O'Rourke

Computer Science: Faculty Publications

The subquadratic algorithm of Kapoor for finding shortest paths on a polyhedron is described.


Computational Geometry Column 33, Joseph O'Rourke Jun 1998

Computational Geometry Column 33, Joseph O'Rourke

Computer Science: Faculty Publications

Several recent SIGGRAPH papers on surface simplification are described.


Computational Geometry Column 34, Pankaj K. Agarwal, Joseph O'Rourke Jan 1998

Computational Geometry Column 34, Pankaj K. Agarwal, Joseph O'Rourke

Computer Science: Faculty Publications

Problems presented at the open-problem session of the 14th Annual ACM Symposium on Computational Geometry are listed.


Computational Geometry Column 32, Joseph O'Rourke Oct 1997

Computational Geometry Column 32, Joseph O'Rourke

Computer Science: Faculty Publications

The proof of Dey's new k-set bound is illustrated.


Shadow Casting Phenomena At Newgrange, Frank Prendergast Jan 1991

Shadow Casting Phenomena At Newgrange, Frank Prendergast

Articles

A digital model of the Newgrange passage tomb and surrounding ring of monoliths known as the Great Circle is used to investigate sunrise shadow casting phenomena at the monument. Diurnal variation in shadow directions and lengths are analysed for their potential use in the Bronze Age to indicate the passage of seasonal time. Computer-aided simulations are developed from a photogrammetric survey to accurately show how three of the largest monoliths, located closest to the tomb entrance and archaeologically coded GC1, GC-1 and GC-2, cast their shadows onto the vertical face of the entrance kerbstone, coded K1. The phenomena occur at …