We consider the problem of enumerating triangulations of n points in the plane in general position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in O(log log n) time per triangulation. It improves the previous bound by almost linear factor. Keywords: Triangulations; Enumeration; Reverse Search 1

Abstract. Perfect, partial, and approximate symmetries are pervasive in 3D surface meshes of real-world objects. However, current digital geometry processing algorithms generally ignore them, instead focusing on local shape features and differential surface properties. This paper investigates how detection of large-scale symmetries can be used to guide processing of 3D meshes. It investigates a framework for mesh processing that includes steps for symmetrization (applying a warp to make a surface more symmetric) and symmetric remeshing (approximating a surface with a mesh having symmetric topology). These steps can be used to enhance the symmetries of a mesh, to decompose a mesh into its symmetric parts and asymmetric residuals, and to establish correspondences between symmetric mesh features. Applications are demonstrated for modeling, beautification, and simplification of nearly symmetric surfaces. Key words: symmetry analysis, mesh processing 1

Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every n-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two n-vertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1 (n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, 3 and there exist triangulations with a maximum simultaneous flip of 6 (n − 2) edges. 7

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.

This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of non-crossing straight edges joining them. A polyhedral surface is a three-dimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a well-defined way of transforming one instance of a struc-ture to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unit-length edges and an-gles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.

Abstract. We develop combinatorial methods for computing the rotation distance between binary trees, i.e., equivalently, the flip distance between triangulations of a polygon. As an application, we prove that, for each n, there exist size n trees at distance 2n − O ( √ n). If T, T ′ are finite binary rooted trees, one says that T ′ is obtained from T by one rotation if T ′ coincides with T except in the neighbourhood of some inner node where the branching patterns respectively are

In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination binary tree must also be degenerate, of a more restricted nature. Previous work on rotation distance has focused on approximation algorithms. Our algorithm is the only known non-trivial polynomial time algorithm for exact rotation distance between special cases of binary trees. 1.

Let S be a set of n points in the plane and let be the set of all crossing-free spanning trees of S. We show that any two trees in can be transformed into each other by O(n ) local and constant-size edge slide operations. No polynomial upper bound for this task has been known, but in [1] a bound of O(n log n) operations was conjectured. Key words: crossing-free spanning tree, local transformation, edge slide 1.

Three-dimensional metamorphosis is a powerful technique to produce a 3D shape transformation between two or more existing models. In this paper, we propose a novel 3D morphing technique that avoids creating a merged embedding that contains the faces, edges, and vertices of two given embeddings. This novel 3D morphing technique dynamically adds or removes vertices to gradually transform the connectivity of 3D polyhedrons from a source model into a target model and simultaneously creates the intermediate shapes. In addition, a priority control function provides the animators with control of arising or dissolving of input models' features in a morphing sequence. This is a useful tool to control a morphing sequence more easily and flexibly. Several examples of aesthetically pleasing morphs are demonstrated using the proposed method.