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Joseph O'Rourke

Publications and source records attributed to Joseph O'Rourke.

At least 19 recordsLinked to original sources
arXiv2026

Overlapping Unfoldings of Cones and Convex Polyhedra

Research on Dürer's problem focuses on edge unfoldings of convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t. We have two main results. The first is that, if we allow unfolding cuts that do not follow polyhedron edges, then there is a convex polyhedron that can unfold with overlap of any given thickness. The second result is that for any given thickness, there is a convex polyhedron with an edge unfolding that overlaps to that thickness.

cs.CG↗
arXiv2026

Hamiltonian Quasigeodesics yield Nets

This note establishes that every polyhedron that has a Hamiltonian quasigeodesic can be edge-unfolded to a net, and shows that the class of such polyhedra is infinite.

cs.CG↗
arXiv2026

Prismatoid Band-Unfolding Revisited

It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.

cs.CG↗
arXiv2026

Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.

cs.CG↗
arXiv2025

Super Guarding and Dark Rays in Art Galleries

We explore an Art Gallery variant where each point of a polygon must be seen by k guards, and guards cannot see through other guards. Surprisingly, even covering convex polygons under this variant is not straightforward. For example, covering every point in a triangle k=4 times (a 4-cover) requires 5 guards, and achieving a 10-cover requires 12 guards. Our main result is tight bounds on k-covering a convex polygon of n vertices, for all k and n. The proofs of both upper and lower bounds are nontrivial. We also obtain bounds for simple polygons, leaving tight bounds an open problem.

cs.CG↗
arXiv2025

Deltahedral Domes over Equiangular Polygons

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths.

math.MG↗
arXiv2025

Skeletal Cut Loci on Convex Polyhedra

On a convex polyhedron P, the cut locus C(x) with respect to a point x is a tree of geodesic segments (shortest paths) on P that includes every vertex. We say that P has a skeletal cut locus if there is some x in P such that C(x) subset Sk(P), where Sk(P) is the 1-skeleton of P. At a first glance, there seems to be very little relation between the cut locus and the 1-skeleton, as the first one is an intrinsic geometry notion, and the second one specifies the combinatorics of P. In this paper we study skeletal cut loci, obtaining four main results. First, given any combinatorial tree T, there exists a convex polyhedron P and a point x in P with a cut locus that lies in Sk(P), and whose combinatorics match T. Second, any (non-degenerate) polyhedron P has at most a finite number of points x for which C(x) subset Sk(P). Third, we show that almost all polyhedra have no skeletal cut locus. Fourth, we provide a combinatorial restriction to the existence of skeletal cut loci. Because the source unfolding of P with respect to x is always a non-overlapping net for P, and because the boundary of the source unfolding is the (unfolded) cut locus, source unfoldings of polyhedra with skeletal cut loci are edge-unfoldings, and moreover "blooming," avoiding self-intersection during an unfolding process. We also explore partially skeletal cut loci, leading to partial edge-unfoldings; i.e., unfoldings obtained by cutting along some polyhedron edges and cutting some non-edges.

cs.CG↗
arXiv2025

Quasigeodesics on the Cube

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le π$ surface to each side at every point. In contrast, a geodesic has exactly $π$ to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly $15$ simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail $15$ simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

math.MG↗
arXiv2023

Every Combinatorial Polyhedron Can Unfold with Overlap

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 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.

cs.CG↗
arXiv2023

Synergies between Venus & Exoplanetary Observations

In this chapter we examine how our knowledge of present day Venus can inform terrestrial exoplanetary science and how exoplanetary science can inform our study of Venus. In a superficial way the contrasts in knowledge appear stark. We have been looking at Venus for millennia and studying it via telescopic observations for centuries. Spacecraft observations began with Mariner 2 in 1962 when we confirmed that Venus was a hothouse planet, rather than the tropical paradise science fiction pictured. As long as our level of exploration and understanding of Venus remains far below that of Mars, major questions will endure. On the other hand, exoplanetary science has grown leaps and bounds since the discovery of Pegasus 51b in 1995, not too long after the golden years of Venus spacecraft missions came to an end with the Magellan Mission in 1994. Multi-million to billion dollar/euro exoplanet focused spacecraft missions such as JWST, and its successors will be flown in the coming decades. At the same time, excitement about Venus exploration is blooming again with a number of confirmed and proposed missions in the coming decades from India, Russia, Japan, the European Space Agency and the National Aeronautics and Space Administration. In this chapter, we review what is known and what we may discover tomorrow in complementary studies of Venus and its exoplanetary cousins.

astro-ph.EP↗
arXiv2022

Reshaping Convex Polyhedra

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 geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P, and to continuously folding P onto Q. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.

math.MG↗
arXiv2022

Simple Closed Quasigeodesics on Tetrahedra

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, and it is only recently that a finite algorithm has been proposed. Here we identify three simple closed quasigeodesics on any tetrahedron: at least one through 1 vertex, at least one through 2 vertices, and at least one through 3 vertices. The only exception is that isosceles tetrahedra have simple closed geodesics but do not have a 1-vertex quasigeodesic. We also identify an infinite class of tetrahedra that each have at least 34 simple closed quasigeodesics.

math.MG↗
arXiv2022

Every Tetrahedron has a 3-vertex Quasigeodesic

We prove that every tetrahedron T has a simple, closed quasigeodesic that passes through three vertices of T. Equivalently, every T has a face whose "exterior angles" are at most pi.

math.MG↗
arXiv2021

Cut Locus Realizations on Convex Polyhedra

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 the star unfolding of P, Alexandrov's gluing theorem, and a cut-locus partition lemma. The construction of P from T is surprisingly simple.

cs.CG↗
arXiv2021

Edge-Unfolding Nearly Flat Convex Caps

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 phi < Phi with the z-axis orthogonal to the halfspace bounding plane. The size of Phi depends on the acuteness gap alpha: if every triangle angle is at most pi/2-alpha, then Phi ~= 0.36 sqrt(alpha) suffices; e.g., for alpha ~= 3deg, Phi = 5deg. 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.

cs.CG↗
arXiv2020

Tailoring for Every Body: Reshaping Convex Polyhedra

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 slices with planes, then Q can be tailored from P. And there is a sense in which tailoring is finer than sculpting in that P may be tailored to polyhedra that are not achievable by sculpting P. It is an easy corollary that, if S is the surface of any convex body, then any convex polyhedron P may be tailored to approximate a homothet of S as closely as desired. So P can be "whittled" to e.g., a sphere S. Another main result achieves the same reshaping, but by excising more complicated shapes we call "crests," still each enclosing one vertex. Reversing either digon-tailoring or crest-tailoring leads to proofs that any Q inside P can be enlarged to P by cutting Q and inserting and sealing surface patches. One surprising corollary of these results is that, for Q a subset of P, we can cut-up Q into pieces and paste them non-overlapping onto an isometric subset of P. This can be viewed as a form of "unfolding" Q onto P. All our proofs are constructive, and lead to polynomial-time algorithms.

math.MG↗
arXiv2020

Some Polycubes Have No Edge Zipper Unfolding

It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no *edge zipper unfolding* where the cut edges are further restricted to form a path.

cs.CG↗