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

Unfolding Convex Polyhedra via Radially Monotone Cut Trees

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, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does not provably find a radially monotone cut tree, it in fact does find such a tree with high frequency, and after cutting unfolds without overlap. This performance of a greedy algorithm leads to the conjecture that spherical polyhedra always have a radially monotone cut tree and unfold without overlap.Comment: 41 pages, 39 figures. V2 updated to cite in an addendum work on "self-approaching curves.

On Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points

The non-convex polyhedron constructed by Chazelle, known as the Chazelle polyhedron [4], establishes a quadratic lower bound on the minimum number of convex pieces for the 3d polyhedron partitioning problem. In this paper, we study the problem of tetrahedralising the Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. The goal of this study is to gain more knowledge about the family of 3d indecomposable polyhedra which needs additional points, so-called Steiner points, to be tetrahedralised. The requirement of only using interior Steiner points for the Chazelle polyhedron is extremely challenging. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d non-convex polyhedron whose vertices are all in the two slightly shifted saddle surfaces which are used to construct the Chazelle polyhedron. We call it the reduced Chazelle polyhedron. It is an indecomposable polyhedron. We then give a set of (N + 1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N + 1) vertices. The proof is done by transforming a 3d tetrahedralisation problem into a 2d edge flip problem. In particular, we design an edge splitting and flipping algorithm and prove that it gives to a tetrahedralisation of the reduced Chazelle polyhedron

Reconstructing Geometric Structures from Combinatorial and Metric Information

In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable. Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets. Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding

An extended MMP algorithm: wavefront and cut-locus on a convex polyhedron

In the present paper, we propose a novel generalization of the celebrated MMP algorithm in order to find the wavefront propagation and the cut-locus on a convex polyhedron with an emphasis on actual implementation for instantaneous visualization and numerical computation.Comment: To appear in International Journal of Computational Geometry & Application

An analysis of mixed integer linear sets based on lattice point free convex sets

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form $S:=\{x : \pi_0 \leq \pi^T x \leq \pi_0+1 \}$ called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be a polyhedron. A split set $S$ has max-facet-width equal to one in the sense that $\max\{\pi^T x : x \in S \}-\min\{\pi^T x : x \in S \} \leq 1$. In this paper we consider using general lattice point free rational polyhedra to derive valid cuts for mixed integer linear sets. We say that lattice point free polyhedra with max-facet-width equal to $w$ have width size $w$. A split cut of width size $w$ is then a valid inequality whose validity follows from a lattice point free rational polyhedron of width size $w$. The $w$-th split closure is the set obtained by adding all valid inequalities of width size at most $w$. Our main result is a sufficient condition for the addition of a family of rational inequalities to result in a polyhedral relaxation. We then show that a corollary is that the $w$-th split closure is a polyhedron. Given this result, a natural question is which width size $w^*$ is required to design a finite cutting plane proof for the validity of an inequality. Specifically, for this value $w^*$, a finite cutting plane proof exists that uses lattice point free rational polyhedra of width size at most $w^*$, but no finite cutting plane proof that only uses lattice point free rational polyhedra of width size smaller than $w^*$. We characterize $w^*$ based on the faces of the linear relaxation