Finding Closed Quasigeodesics on Convex Polyhedra

A closed quasigeodesic is a closed loop on the surface of a polyhedron with at most $180^\circ$ of surface on both sides at all points; such loops can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron has at least three (non-self-intersecting) closed quasigeodesics, but the proof relies on a nonconstructive topological argument. We present the first finite algorithm to find a closed quasigeodesic on a given convex polyhedron, which is the first positive progress on a 1990 open problem by O'Rourke and Wyman. The algorithm's running time is pseudopolynomial, namely $O\left({n^2 \over \varepsilon^2} {L \over \ell} b\right)$ time, where $\varepsilon$ is the minimum curvature of a vertex, $L$ is the length of the longest edge, $\ell$ is the smallest distance within a face between a vertex and a nonincident edge (minimum feature size of any face), and $b$ is the maximum number of bits of an integer in a constant-size radical expression of a real number representing the polyhedron. We take special care with the model of computation, introducing the $O(1)$-expression RAM and showing that it can be implemented in the standard word RAM.Comment: 18 pages, 11 figures. Revised version of paper from SoCG 202

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

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. v2 improves the description of cut locus, and adds references. v3 improves two figures and their captions. New version v4 offers a completely different proof of non-overlap in the quasigeodesic loop case, and contains several other substantive improvements. This version is 23 pages long, with 15 figure

Source Unfoldings of Convex Polyhedra via Certain Closed Curves

Abstract. 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 unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygo

Source Unfoldings of Convex Polyhedra via Certain Closed Curves

Abstract. 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 unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygo

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. 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.Comment: Research monograph. 234 pages, 105 figures, 55 references. arXiv admin note: text overlap with arXiv:2008.0175

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry

The Star Unfolding from a Geodesic Curve

An unfolding of a polyhedron P is obtained by `cutting' the surface of P in such a way that it can be flattened into the plane into a single polygon. For most practical and theoretic applications, it is desirable for an algorithm to produce an unfolding which is simple, that is, non-overlapping. Currently, two methods for unfolding which guarantee non-overlap for convex polyhedra are known, the source unfolding, and the star}unfolding. Both methods involve computing shortest paths from a single source point on the polyhedron's surface. In this thesis, we attempt to prove non-overlap of a variant called the geodesic star unfolding. This unfolding, much like the star unfolding, is computed by cutting shortest paths from each vertex to λ, a geodesic curve on the surface of a convex polyhedron P, and also cutting λ itself. Non-overlap of this case was conjectured by Demaine and Lubiw (2011). We are unsuccessful in completely proving non-overlap, though we present a number of partial results, and discuss some areas for future study. We first develop a new proof for non-overlap of the star unfolding from a point. The original proof of non-overlap was given by Aronov and O'Rourke (2009). This new proof uses a partitioning of the unfolding around the ridge tree. Each edge of the ridge tree serves as a base edge to a pair of congruent triangles; in this way, the whole unfolding is decomposed into these pairs which are called kites. We prove non-overlap by showing that pairwise, no two kites in the unfolding overlap each other, by a method which bounds the surface angle of the source images to either side of any path through the ridge tree. In addition to its simplicity compared to the previous proof, this new method easily generalizes to prove non-overlap for some cases of the star unfolding from geodesic curves. Specifically, we show non-overlap for two classes of geodesic curves, geodesic loops, and fully-extended S-shaped geodesics, by showing that the surface angle of the source images in those two cases are bounded. We also investigate a class of curves called fully-extended C-shaped geodesics for which the proof cannot hold directly. We show some specific cases where we are able to create a supplementary proof to show non-overlap, though non-overlap for the class as a whole remains unproven