Transforming triangulations on non planar-surfaces

We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM Journal on Discrete Mathematics. Keywords: Graph of triangulations, triangulations on surfaces, triangulations of polygons, edge fli

Transforming triangulations on non planar-surfaces

We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinde

Transforming Triangulations on Nonplanar Surfaces

We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder

Ear-clipping Based Algorithms of Generating High-quality Polygon Triangulation

A basic and an improved ear clipping based algorithm for triangulating simple polygons and polygons with holes are presented. In the basic version, the ear with smallest interior angle is always selected to be cut in order to create fewer sliver triangles. To reduce sliver triangles in further, a bound of angle is set to determine whether a newly formed triangle has sharp angles, and edge swapping is accepted when the triangle is sharp. To apply the two algorithms on polygons with holes, "Bridge" edges are created to transform a polygon with holes to a degenerate polygon which can be triangulated by the two algorithms. Applications show that the basic algorithm can avoid creating sliver triangles and obtain better triangulations than the traditional ear clipping algorithm, and the improved algorithm can in further reduce sliver triangles effectively. Both of the algorithms run in O(n2) time and O(n) space.Comment: Proceedings of the 2012 International Conference on Information Technology and Software Engineering Lecture Notes in Electrical Engineering Volume 212, 2013, pp 979-98

Flip Distance Between Two Triangulations of a Point-Set is NP-complete

Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane

Flip Paths Between Lattice Triangulations

Diagonal flip paths between triangulations have been studied in the combinatorial setting for nearly a century. One application of flip paths to Euclidean distance geometry and Moebius geometry is a recent, simple, constructive proof by Connelly and Gortler of the Koebe-Andreev-Thurston circle packing theorem that relies on the existence of a flip path between any two triangulation graphs. More generally, length and other structural quantities on minimum (length) flip paths are metrics on the space of triangulations. In the geometric setting, finding a minimum flip path between two triangulations is NP-complete. However, for two lattice triangulations, used to model electron spin systems, Eppstein and Caputo et al. gave algorithms running in $O\left(n^2\right)$ time, where $n$ is the number of points in the point-set. Their algorithms apply to constrained flip paths that ensure a set of \emph{constraint} edges are present in every triangulation along the path. We reformulate the problem and provide an algorithm that runs in $O\left(n^{\frac{3}{2}}\right)$ time. In fact, for a large, natural class of inputs, the bound is tight, i.e., our algorithm runs in time linear in the length of this output flip path. Our results rely on structural elucidation of minimum flip paths. Specifically, for any two lattice triangulations, we use Farey sequences to construct a partially-ordered sets of flips, called a minimum flip \emph{plan}, whose linear-orderings are minimum flip paths between them. To prove this, we characterize a minimum flip plan that starts from an equilateral lattice triangulation - i.e., a lattice triangulation whose edges are all unit-length - and \emph{forces a point-pair to become an edge}. To the best of our knowledge, our results are the first to exploit Farey sequences for elucidating the structure of flip paths between lattice triangulations.Comment: 24 pages (33 with appendices), 8 figure

Survey of two-dimensional acute triangulations

AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual

Topological mechanics of origami and kirigami

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom.Comment: 5 pages, 3 figures + ~5 pages S