# Joseph O'RourkeSmith College

Joseph O'Rourke

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306

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## Publications

Publications (306)

Pogorelov proved in 1949 that every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly a π surface angle to either side at each point, a quasigeodesic has at most a π surface angle to either side at each point. Pogorelov’s existence proof did not suggest a way to identify the three quasigeodesics, and...

Pogorelov proved in 1949 that every every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly pi surface angle to either side at each point, a quasigeodesic has at most pi surface angle to either side at each point. Pogorelov's existence proof did not suggest a way to identify the three quasigeodesics,...

We present nonoverlapping general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n-pointed star polygon whose triangular points have base angle \(\alpha \). Specifically, we construct general unfoldings when \(n \in \{3,4,5,6,8,9,10,12\}\) (no matter the value of...

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geod...

We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuum of polyhedra P each realizing T for some x on P. Three main tools in the proof are properties of t...

Given any two convex polyhedra P and Q, we prove as one of our main results that the surface of P can be reshaped to a homothet of Q by a finite sequence of "tailoring" steps. Each tailoring excises a digon surrounding a single vertex and sutures the digon closed. One phrasing of this result is that, if Q can be "sculpted" from P by a series of sli...

Starting with the unsolved "D\"urer's problem" of edge-unfolding a convex polyhedron to a net, we specialize and generalize (a) the types of cuts permitted, and (b) the polyhedra shapes, to highlight both advances established and which problems remain open.

It is unknown whether or not every polycube has an edge-unfolding. A polycube is an object constructed by gluing cubes face-to-face. An edge-unfolding cuts edges on the surface and unfolds it to a net, a non-overlapping polygon in the plane. Here we explore the more restricted edge-unzippings where the cut edges form a path. We construct a polycube...

An unzipping of a polyhedron P is a cut-path through its vertices that unfolds P to a non-overlapping shape in the plane. It is an open problem to decide if every convex P has an unzipping. Here we show that there are nearly flat convex caps that have no unzipping. A convex cap is a "top" portion of a convex polyhedron; it has a boundary, i.e., it...

We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. Our main result is a linear-time algorithm for deciding whether a polygonal curve is threadable, and if so, finding a sequence of rigid motions to thread...

This addendum to [O'R17] establishes that a nearly flat acutely triangulated convex cap in the sense of that paper can be edge-unfolded even if closed to a polyhedron by adding the convex polygonal base under the cap.

We show that every orthogonal polyhedron of genus g≤2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call t...

This paper proves a conjecture from [LO17]: A nearly flat, acutely triangulated convex cap C has an edge-unfolding to a non-overlapping polygon in the plane. "Nearly flat" means that every face normal forms a sufficiently small angle with the z-axis. Although the result is not surprising, the proof relies on some recently developed concepts, angle-...

We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted span...

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar...

We show that the hypercube has a face-unfolding that tiles space, and that
unfolding has an edge-unfolding that tiles the plane. So the hypercube is a
"dimension-descending tiler." We also show that the hypercube cross unfolding
made famous by Dali tiles space, but we leave open the question of whether or
not it has an edge-unfolding that tiles the...

The notion of a spiral unfolding of a convex polyhedron, resulting by
flattening a special type of Hamiltonian cut-path, is explored. The Platonic
and Archimedian solids all have nonoverlapping spiral unfoldings, although
among generic polyhedra, overlap is more the rule than the exception. The
structure of spiral unfoldings is investigated, primar...

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in -space. It is straightforward to show that any box graph in the plane has an admissible coloring with three colors, and that any box graph in -space has an admissible coloring with colors. We show that for box graphs in -space, if the length...

We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhed...

We introduce the problem of draining water (or balls representing water drops) out of a punctured polygon (or a polyhedron) by rotating the shape. For 2D polygons, we obtain combinatorial bounds on the number of holes needed, both for arbitrary polygons and for special classes of polygons. We detail an O(n2logn) algorithm that finds the minimum num...

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each...

It is an open problem, posed in \cite{SoCG}, to determine the minimal $k$
such that an open flexible $k$-chain can interlock with a flexible 2-chain. It
was first established in \cite{GLOSZ} that there is an open 16-chain in a
trapezoid frame that achieves interlocking. This was subsequently improved in
\cite{GLOZ} to establish interlocking between...

For a set of points in the plane and a fixed integer k > 0, the Yao graph Y_k
partitions the space around each point into k equiangular cones, and connects
each point to a nearest neighbor in each cone. With the exception of Y_5, it is
known for all Yao graphs whether or not they are geometric spanners. In this
paper we resolve this gap and show th...

In 1525 the German painter and thinker Albrect Dürer published his masterwork on geometry, whose title translates as “On Teaching Measurement with a Compass and Straightedge.”

As a pedagogical exercise, we derive the shape of a particularly elegant
pop-up card design, and show that it connects to a classically studied plane
curve that is (among other interpretations) a caustic of a circle.

We address the unsolved problem of unfolding prismatoids in a new context,
viewing a "topless prismatoid" as a convex patch---a polyhedral subset of the
surface of a convex polyhedron homeomorphic to a disk. We show that several
natural strategies for unfolding a prismatoid can fail, but obtain a positive
result for "petal unfolding" topless prisma...

We extend the notion of a source unfolding of a convex polyhedron P to be
based on a closed polygonal curve Q in a particular class rather than based on
a point. The class requires that Q "lives on a cone" to both sides; it includes
simple, closed quasigeodesics. Cutting a particular subset of the cut locus of
Q (in P) leads to a non-overlapping un...

The first Computational Geometry Columns were all printed in ACM SIGACT News; some also appeared in Computer Graphics, the newsletter for ACM SIGGRAPH. Starting with this eleventh column, it will be printed in both SIGACT News and the International Journal of Computational Geometry & Applications. I have collected the first ten columns in a Smith T...

Let the centers of a finite number of disjoint, closed disks be pinned to the plane, but with each free to rotate about its center. Given an arrangement of such disks with each labeled + or −, we inves-tigate the question of whether they can be all wrapped by a single loop of string so that, when the string is taut and circulates, it rotates by fri...

It is shown that there are examples of distinct polyhedra, each with a
Hamiltonian path of edges, which when cut, unfolds the surfaces to a common
net. In particular, it is established for infinite classes of triples of
tetrahedra.

Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two.Discrete and Computa...

Let C be a simple, closed, directed curve on the surface of a convex
polyhedron P. We identify several classes of curves C that "live on a cone," in
the sense that C and a neighborhood to one side may be isometrically embedded
on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the
image of) C; we also prove that each point...

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.

In this paper, we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand/contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2 × 2 × 2 modules. We respect cert...

We address the question: How many edge guards are needed to guard an orthogonal polyhedron of e edges, r of which are reflex? It was previously established [3] that e/12 are sometimes necessary and e/6 always suffice. In contrast to the closed edge guardsused for these bounds, we introduce a new model, open edge guards (excluding the endpoints of t...

We establish that certain classes of simple, closed, polygonal curves on the surface of a convex polyhedron develop in the plane without overlap. Our primary proof technique shows that such curves “live on a cone,” and then develops the curves by cutting the cone along a “generator” and flattening the cone in the plane. The conical existence result...

What do proteins and pop-up cards have in common? How is opening a grocery bag different from opening a gift box? How can you cut out the letters for a whole word all at once with one straight scissors cut? How many ways are there to flatten a cube? With the help of 200 colour figures, author Joseph O'Rourke explains these fascinating folding probl...

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...

We examine the problem of pushing all the points of a planar region into one point using parallel sweeps of an innite line, minimizing the sum of the lengths of the sweep vectors. We characterize the optimal 2-sweeps of triangles, and provide a linear-time algorithm for convex polygons.

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper...

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n
2) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and “twangs,” which
walk between weakly simple “polygonal wraps” of S. These moves show promise to serve as a basis for generating random p...

We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier. Comment: 6 pages, 1 figure

We show that there is a straightforward algorithm to determine if the
polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate
flat polyhedron, and to reconstruct it from the gluing instructions. The
algorithm runs in O(n^3) time for polygons whose gluings are specified by n
labels.

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron ℘ to a simple (nonoverlapping)
planar polygon: cut along one shortest path from each vertex of ℘ to Q, and cut all but one segment of Q.
KeywordsUnfolding-Star unfolding...

We show that the Yao graph Y
4 in the L
2 metric is a spanner with stretch factor 8Ö2(29+23Ö2)8\sqrt{2}(29+23\sqrt{2}).

We prove that Y_6 is a spanner. Y_6 is the Yao graph on a set of planar points, which has an edge from each point x to a closest point y within each of the six angular cones of 60 deg surrounding x. Comment: 12 pages, 11 figures, 2 references, unpublished. A subcase of Case(2) of Theorem 1 was overlooked, and I withdraw the paper until that gap is...

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch
factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in
the Linf metric is a planar spanner with stretch factor 8.

In this paper we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand/contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2 x 2 x 2 modules. We respect certa...

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. Comment: 13 pages, 6 figures

The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales
it claims that the complement of a chord that uses half the pitches of a scale is homometric to—i.e., has the same interval
structure as—the original chord. In terms of onsets it claims that the complement of a rhythm with the s...

The Yao graph for k=4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners. Comment: 7 pages, 6 figures

We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.

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) atom operations (expand, contract, attach, detach) and is performed in O(n) parallel steps. This improves on p...

Can a simple spherical polygon always be triangulated? The answer depends on the definitions of "polygon" and "triangulate".
Define a spherical polygon to be a simple, closed curve on a sphere S composed of a finite number of great circle arcs (also known as geodesic arcs) meeting at vertices. Can every spherical polygon be triangulated? Figure 1 s...

A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H.The highway hullH(S,H) of a point set S is the minimal...

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and...

We present a novel approach to morph between two isometric poses of the same
non-rigid object given as triangular meshes. We model the morphs as linear
interpolations in a suitable shape space $\mathcal{S}$. For triangulated 3D
polygons, we prove that interpolating linearly in this shape space corresponds
to the most isometric morph in $\mathbb{R}^...

Two long-open problems have been solved: (1) every suciently large planar point set in general position contains the vertices of an empty hexagon; (2) every finite collection of polygons of equal area have a common hinged dissection.

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, 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 investigate a question initiated in the work of T. Sibley and S. Wagon [Am. Math. Mon. 107, No. 3, 251–253 (2000; Zbl 0983.52014)], who proved that 3 colors suffice to color any collection of 2D parallelograms glued edge-to-edge. Their proof relied on the existence of an “elbow” parallelogram. We explore the existence of analogous “corner” paral...

A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel p...

We introduce the problem of draining water (or balls repre- senting water drops) out of a punctured polygon (or a poly- hedron) by rotating the shape. For 2D polygons, we obtain combinatorial bounds on the number of holes needed, both for arbitrary polygons and for special classes of polygons. We detail an O(n2 logn) algorithm that finds the minimu...

Homometric rhythms (chords) are those with the same histogram or multiset of intervals (distances). The purpose of this note is threefold. First, to point out the potential importance of isospectral vertices in a pair of homometric rhythms. Second, to establish a method ("pumping") for generating an infinite sequence of homometric rhythms that incl...

We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space S. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in R3. We extend this shape...

Every simple planar polygon can undergo only a finite number of pocket flips before becoming convex. Since Erd˝ os posed this finiteness as an open problem in 1935, several independent purported proofs have been published. However, we uncover a plethora of errors, gaps, and omissions in these arguments, leaving only two proofs without flaws and no...

Recent progress is described on the unsolved problem of unfold- ing the surface of an orthogonal polyhedron to a single non-overlapping planar piece by cutting edges of the polyhedron. Although this is in general not possible, partitioning the faces into the natural vertex-grid may render it al- ways achievable. Advances that have been made on vari...

Soss proved that it is NP-hard to find the maximum 2D span of a fixed-angle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixed-angle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that three special cases of particular relevance to the protein...

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 2×2×22×2×2 meta-modules. The reconfiguration involves a total of O(n)O(n) atomic operations (expand, contract, attach, detach) and is performed in O(n)O(n) parallel steps. This...

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the pri...

The problem of wireless localization asks to place and orient stations in the plane, each of which broadcasts a unique key within a fixed angular range, so that each point in the plane can determine whether it is inside or outside a given polygonal region. The primary goal is to minimize the number of stations. In this paper we establish a lower bo...

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n^2) ``moves'' between simple polygons. Each move is composed of a sequence of atomic moves called ``stretches'' and ``twangs''. These atomic moves walk between weakly simple ``polygonal wraps'' of S. These moves show promise to serve as a basis fo...

This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap's faces are quadrilaterals, with vertices over an underlying integer lattice, and such that the cap convexity is ``radially monotone,'' a type of smoothnes...

It is shown that every 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.

The concept of pointed pseudo-triangulations is defined and a few of its
applications described.

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q. Comment: 10 pages, 7 figures. v...

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 algorithm cuts along edges of a 4x5x1 refinement of the vertex grid.

The new algorithm of Bobenko and Izmestiev for reconstructing the unique polyhedron determined by given gluings of polygons is described.

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we
prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at righ...

We show that cutting shortest paths from every vertex of a convex polyhedron to a simple closed quasigeodesic, and cutting all but a short segment of the quasigeodesic, unfolds the surface to a planar simple poly- gon.

Did you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even ‘sign your name’? Or that a ‘Latin cross’ unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past de...

We prove that an infinite class of convex polyhedra, pro- duced by restricted vertex truncations, always unfold with- out overlap. The class includes the "domes," providing a simpler proof that these unfold without overlap.

A new class of art gallery-like problems inspired by wireless localization is discussed.

A remarkable theorem is described: \It is possible to tile the plane with non- overlapping squares using exactly one square of each integral dimension." Thus, one can \square the plane." More than thirty years ago, Solomon Golomb posed (Gol75) the question of whether or not the innite plane could be tiled using one copy of each square of integer si...

It has recently been established by Below, De Loera, and Richter-Gebert
that finding a minimum size (or even just a small) triangulation of a
convex polyhedron is NP-complete. Their 3SAT-reduction proof is
discussed.