Subquadratic nonobtuse triangulation of convex polygons

A convex polygon with n sides can be triangulated by O(n^1.85) triangles, without any obtuse angles. The construction uses a novel form of geometric divide and conquer

An Algorithm for Triangulating 3D Polygons

In this thesis, we present an algorithm for obtaining a triangulation of multiple, non-planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non-manifold outputs for two and more input polygons without compromising opti- mality. For better performance on real-world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality

A Contribution to Triangulation Algorithms for Simple Polygons

Decomposing simple polygon into simpler components is one of the basic tasks in computational geometry and its applications. The most important simple polygon decomposition is triangulation. The known algorithms for polygon triangulation can be classified into three groups: algorithms based on diagonal inserting, algorithms based on Delaunay triangulation, and the algorithms using Steiner points. The paper briefly explains the most popular algorithms from each group and summarizes the common features of the groups. After that four algorithms based on diagonals insertion are tested: a recursive diagonal inserting algorithm, an ear cutting algorithm, Kong’s Graham scan algorithm, and Seidel’s randomized incremental algorithm. An analysis concerning speed, the quality of the output triangles and the ability to handle holes is done at the end

Simple and Robust Boolean Operations for Triangulated Surfaces

Boolean operations of geometric models is an essential issue in computational geometry. In this paper, we develop a simple and robust approach to perform Boolean operations on closed and open triangulated surfaces. Our method mainly has two stages: (1) We firstly find out candidate intersected-triangles pairs based on Octree and then compute the inter-section lines for all pairs of triangles with parallel algorithm; (2) We form closed or open intersection-loops, sub-surfaces and sub-blocks quite robustly only according to the cleared and updated topology of meshes while without coordinate computations for geometric enti-ties. A novel technique instead of inside/outside classification is also proposed to distinguish the resulting union, subtraction and intersection. Several examples have been given to illus-trate the effectiveness of our approach.Comment: Novel method for determining Union, Subtraction and Intersectio

The Gromov Norm of the Product of Two Surfaces

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.Comment: The journal version contains an error that invalidates one direction of the main theorem. The present version contains an erratum, at the end, explaining thi

Delaunay-restricted Optimal Triangulation of 3D Polygons

Triangulation of 3D polygons is a well studied topic of research. Existing methods for finding triangulations that minimize given metrics (e.g., sum of triangle areas or dihedral angles) run in a costly O(n4) time [BS95,BDE96], while the triangulations are not guaranteed to be free of intersections. To address these limitations, we restrict our search to the space of triangles in the Delaunay tetrahedralization of the polygon. The restriction allows us to reduce the running time down to O(n2) in practice (O(n3) worst case) while guaranteeing that the solutions are intersection free. We demonstrate experimentally that the reduced search space is not overly restricted. In particular, triangulations restricted to this space usually exist for practical inputs, and the optimal triangulation in this space approximates well the optimal triangulation of the polygon. This makes our algorithms a practical solution when working with real world data