Papers by Joseph O'Rourke
arXiv (Cornell University), Jul 4, 2017
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in... more The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R 3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. "Nearly flat" means that every outer face normal forms a sufficiently small angle φ < Φ with theẑ-axis orthogonal to the halfspace bounding plane. The size of Φ depends on the acuteness gap α: if every triangle angle is at most π/2−α, then Φ ≈ 0.36 √ α suffices; e.g., for α = 3 • , Φ ≈ 5 •. Even if C is closed to a polyhedron by adding the convex polygonal base under C, this polyhedron can be edge-unfolded without overlap. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n 2); a version has been implemented.
arXiv (Cornell University), Jun 10, 2022
This note establishes that every polyhedron that has a Hamiltonian quasigeodesic can be edge-unfo... more This note establishes that every polyhedron that has a Hamiltonian quasigeodesic can be edge-unfolded to a net.
A system capable of analyzing image sequences of human motion is described. The system is structu... more A system capable of analyzing image sequences of human motion is described. The system is structured as a •feedback loop between high and low levels: predictions are made at the semantic level, and verifications are sought at the image level. The domain of human motion lends itself to a model-driven analysis, and the system includes a detailed model of the human body. All information extracted from the image is interpreted through a constraint network based on the structure of the human model. A constraint propagation operator is defined and its theoretical,properties developed. An implementation of this operator is described, and results of the analysis system for a short image sequence are presented.
The need to visualize and interpret human body movement data from experiments and simulations has... more The need to visualize and interpret human body movement data from experiments and simulations has led to the development of a new, computerized, three-dimensional representation for the human body. Based on a skeleton of joints and segments, the model is manipulated by specifying joint positions with respect to arbitrary frames of reference. The external form is modelled as the union of overlapping spheres which define the surface of each segment. The properties of the segment and sphere model include: an ability to utilize any connected portion of the body in order to examine selected movements without computing movements of undesired parts , a naming mechanism for describing parts within a segment, and a collision detection algorithm for finding contacts or illegal intersections of the body with itself or other objects. One of the most attractive features of this model is the simple hidden surface removal algorithm. Since spheres always project onto a plane as disks, a solid, shad...
Lecture Notes in Computer Science, 1989
The center of area of a polygon P is the unique point p* that maximizes the minimum area overlap ... more The center of area of a polygon P is the unique point p* that maximizes the minimum area overlap between P and any halfplane that includes p*. We present several "numerical" algorithms for finding the coordinates of p* for a polygon of n vertices. These algorithms are numerical in the sense that we have been careful to
The Visual Computer, 1994
The center of area of a convex polygon P is the unique point p* that maximizes the minimum area o... more The center of area of a convex polygon P is the unique point p* that maximizes the minimum area overlap between P and any halfplane that includes p*. We show that p* is unique and present two algorithms for its computation. The first is a combinatorial algorithm that runs in time O(n 6 log 2 n). The second is a "numerical" algorithm that runs in time O(GK(n + K)) where K represents the number of desired bits of precision in the output coordinates and G the number of bits used to represent the coordinates of the input polygon vertices. We conclude with a discussion of implementation issues and related results.
International Journal of Computational Geometry & Applications, 2012
We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor [Formula: see tex... more We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor [Formula: see text]. Enroute to this, we also show that the Yao graph [Formula: see text] in the L∞ metric is a plane spanner with stretch factor 8.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 1980
A system capable of analyzing image sequences of human motion is described. The system is structu... more A system capable of analyzing image sequences of human motion is described. The system is structured as a •feedback loop between high and low levels: predictions are made at the semantic level, and verifications are sought at the image level. The domain of human motion lends itself to a model-driven analysis, and the system includes a detailed model of the human body. All information extracted from the image is interpreted through a constraint network based on the structure of the human model. A constraint propagation operator is defined and its theoretical,properties developed. An implementation of this operator is described, and results of the analysis system for a short image sequence are presented.
Computational Geometry, 2003
We prove NP-hardness of a wide class of pushing-block puzzles similar to the classic Sokoban, gen... more We prove NP-hardness of a wide class of pushing-block puzzles similar to the classic Sokoban, generalizing several previous results [5, 6, 9, 10, 15, 17]. The puzzles consist of unit square blocks on an integer lattice; all blocks are movable. The robot may move horizontally and vertically in order to reach a specified goal position. The puzzle variants differ in the number of blocks that the robot can push at once, ranging from at most one (PUSH-1) up to arbitrarily many (PUSH-*). Other variations were introduced to make puzzles more tractable, in which blocks must slide their maximal extent when pushed (PUSHPUSH), and in which the robot's path must not revisit itself (PUSH-X). We prove that all of these puzzles are NP-hard.
Computational Geometry, 2009
In this paper we propose a novel algorithm that, given a source robot S and a target robot T , re... more In this paper we propose a novel algorithm that, given a source robot S and a target robot T , reconfigures S into T. Both S and T are robots composed of n atoms arranged in 2 × 2 × 2 meta-modules. The reconfiguration involves a total of O (n) atomic operations (expand, contract, attach, detach) and is performed in O (n) parallel steps. This improves on previous reconfiguration algorithms [D.
Computational Geometry, 2003
We study collections of linkages in 3-space that are interlocked in the sense that the linkages c... more We study collections of linkages in 3-space that are interlocked in the sense that the linkages cannot be separated without one bar crossing through another. We explore pairs of linkages, one open chain and one closed chain, each with a small number of joints, and determine which can be interlocked. In particular, we show that a triangle and an open 4-chain can interlock, a quadrilateral and an open 3-chain can interlock, but a triangle and an open 3-chain cannot interlock.
arXiv (Cornell University), Dec 30, 2022
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that... more Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result as establishing that every combinatorial polyhedron has a metric realization that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of "reverse"): Is there a combinatorial polyhedron P such that, for every metric realization P in R 3 , and for every spanning cut-tree T , P cut by T unfolds to a net? In this note we prove the answer is no: every combinatorial polyhedron has a realization and a cut-tree that unfolds the polyhedron with overlap.
Eprint Arxiv 1101 0823, Jan 4, 2011
Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex... more Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.
arXiv (Cornell University), Dec 17, 2010
We establish a simple generalization of a known result in the plane. The simplices in any pure si... more We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R d may be colored with d+1 colors so that no two simplices that share a (d−1)-facet have the same color. In R 2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R 3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.
Proceedings of the fourth annual symposium on Computational geometry - SCG '88
Let an arrangement of blue lines in 3-space be fixed, and imagine a movable red line entangled in... more Let an arrangement of blue lines in 3-space be fixed, and imagine a movable red line entangled in the arrangement. We show an O(n•(n)) algorithm for building a data structure that permits enumeration of mutually inaccessible classes of such red lines, where c~(n) is the inverse Ackermann function. The core of the algorithm is a construction of O(n 2) 2-D arrangements of hyperbolas, each in O(n~a(n)) time. The algorithm is applied to stabbing 3-polytopes, enumerating pairwisevisible face pairs, enumerating 2-D projections of convex 4-polytopes, and other problems, resulting in O(n4c~(n)) algorithms in each case.
arXiv (Cornell University), Nov 28, 1999
We prove that a particular pushing-blocks puzzle is intractable in 3D. The puzzle, inspired by th... more 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.
Proceedings of the second annual symposium on Computational geometry - SCG '86, 1986
EIGindy and Avis [EA] considered the problem of determining the visibility polygon from a point i... more EIGindy and Avis [EA] considered the problem of determining the visibility polygon from a point inside a polygon. Their algorithm runs in optimal O(n) time and space, where n is the number of the vertices of the given polygon. Later their result was generalized to visibility polygons from an edge by EIGindy [Eli, and Lee and Lin
It is an open problem, posed in [3], to determine the minimal k such that an open flexible k-chai... more It is an open problem, posed in [3], to determine the minimal k such that an open flexible k-chain can interlock with a flexible 2-chain. It was first established in [5] that there is an open 16-chain in a trapezoid frame that achieves interlocking. This was subsequently improved in [6] to establish interlocking between a 2-chain and an open 11-chain. Here we improve that result once more, establishing interlocking between a 2-chain and a 10-chain. We present arguments that indicate that 10 is likely the minimum.
Corr, Jan 25, 2008
We construct a sequence of convex polyhedra on n vertices with the property that, as n→∞, the fra... more We construct a sequence of convex polyhedra on n vertices with the property that, as n→∞, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.
We address a question posed by Sibley and Wagon. They proved that rhombic Penrose tilings in the ... more We address a question posed by Sibley and Wagon. They proved that rhombic Penrose tilings in the plane can be 3colored, but a key lemma of their proof fails in the natural 3D generalization. In that generalization, an object is built from bricks, each of which is a parallelopiped, and they are glued face-to-face. The question is: How many colors are needed to color the bricks of any such object, with no two face-adjacent bricks receiving the same color? For arbitrary parallelopiped bricks, we prove zonohedra are 4-colorable, and 4 colors are sometimes necessary, by establishing two Sibley conjectures for zonohedra. For orthogonal bricks, we narrow the chromatic number to {3, 4}, and have several results. Any genus-0 object (a "ball") is 2-colorable; any genus-1 object is 3-colorable. For objects of higher genus, we show that if an object's holes are "nonplanar" in a technical sense, then it is 2-colorable regardless of its genus, and for various special cases of planar holes, we can establish 3-colorability. We conjecture that all objects built from orthogonal bricks are 3-colorable. This would imply that the chromatic number does not increase when passing from 2D to 3D.
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Papers by Joseph O'Rourke