Triangulations in CGAL

1998

The Multi-Triangulation (MT) is a general framework for managing the Level-of-Detail in large triangle meshes, which we have introduced in our previous work. In this paper, we describe an efficient implementation of an MT based on vertex decimation. We present general techniques for querying an MT, which are independent of a specific application, and which can be applied for solving problems, such as selective refinement, windowing, point location, and other spatial interference queries. We describe alternative data structures for encoding an MT, which achieve different trade-offs between space and performance. Experimental results are discussed.

1990

Lecture Notes in Computer Science, 2004

In this paper we give an algorithm to generate all biconnected plane triangulations having exactly n vertices including exactly r vertices on the outer face. The algorithm uses O(n) space in total and generates such triangulations without duplications in O(rn) time per triangulation, while the previous best algorithm generates such triangulations in O(r 2 n) time per triangulation.

ACM Communications in Computer Algebra, 2014

The Computational Geometry Algorithms Library (CGAL) is an open source software library that provides industrial and academic users with easy access to reliable implementations of efficient geometric algorithms. Usage. CGAL is used in a diverse range of domains requiring geometric computation such as computer graphics, scientific visualization, computer aided design and modeling, geographic information systems, molecular biology, medical imaging, and many more. Since CGAL provides a wide range of components, we restrict ourselves to mentioning just a few here. As an example application of CGAL, a series of packages are provided which are useful in robotics and automation: Minkowski sums, offset polygons, Boolean operations on curved regions. The high precision of CGAL allows users to solve geometric problems involving motion in restricted environments, such as those arising in assembly planning. The robustness and efficiency of components such as the Delaunay triangulation and mesh construction and manipulation packages makes CGAL attractive for simulations, in particular those involving proteins, particle physics, fluid dynamics, medical modeling, biophysics, geophysics, and astronomy. Indeed, the aforementioned components are largely used in these areas. Some support for manipulations of polynomials and for solving univariate polynomial equations and bivariate polynomial systems is also provided, as well as handling for convex quadratic programs.

Computational Geometry, 2004

A "rooted" plane triangulation is a plane triangulation with one designated vertex on the outer face. A simple algorithm to generate all triconnected rooted plane triangulations with at most n vertices is presented. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm all triconnected rooted plane triangulations having exactly n vertices including exactly r vertices on the outer face in O(r) time per triangulation can be generated without duplicates, while the previous best algorithm generates such triangulations in O(n 2) time per triangulation. All triconnected (non-rooted) plane triangulations having exactly n vertices including exactly r vertices on the outer face can also be generated without duplicates in O(r 2 n) time per triangulation, and all maximal planar graphs can be generated in O(n 3) time per graph.

Computers Graphics, 2006

Spatial sampling methods have acquired great popularity due to the number of applications that need to triangulate portions of space in various dimensions. One limitation of the current techniques is the handling of the final models, which are large, complex and need to register neighborhood relationships explicitly. Additionally, most techniques are limited to Euclidean bidimensional or tridimensional spaces and many do not handle well adaptive refinement. This work presents a novel method for spatial decomposition based on simplicial meshes (the J a 1 triangulation) that is generally defined for Euclidean spaces of any dimension and is intrinsically adaptive. Additionally it offers algebraic mechanisms for the decomposition itself and for definition of neighbrs that allow to recover all the information on the resulting mesh via a set of rules. This way it is possible to balance the cost of storage and manipulation by calculating the needed information instead of storing it. Results additionally show good quality meshes with efficient calculation.

Computers & Graphics, 2006

Spatial sampling methods have acquired great popularity due to the number of applications that need to triangulate portions of space in various dimensions. One limitation of the current techniques is the handling of the final models, which are large, complex and need to register neighborhood relationships explicitly. Additionally, most techniques are limited to Euclidean bidimensional or tridimensional spaces and many do not handle well adaptive refinement. This work presents a novel method for spatial decomposition based on simplicial meshes (the J a 1 triangulation) that is generally defined for Euclidean spaces of any dimension and is intrinsically adaptive. Additionally it offers algebraic mechanisms for the decomposition itself and for definition of neighbrs that allow to recover all the information on the resulting mesh via a set of rules. This way it is possible to balance the cost of storage and manipulation by calculating the needed information instead of storing it. Results additionally show good quality meshes with efficient calculation.

ISPRS International Journal of Geo-Information, 2017

The Delaunay triangulation is the standard choice for building triangulated irregular networks (TINs) to represent terrain surfaces. However, the Delaunay triangulation is based only on the 2D coordinates of the data points, ignoring their elevation. This can affect the quality of the approximating surface. In fact, it has long been recognized that sometimes it may be beneficial to use other, non-Delaunay, criteria that take elevation into account to build TINs. Data-dependent triangulations were introduced decades ago to address this exact issue. However, data-dependent trianguations are rarely used in practice, mostly because the optimization of data-dependent criteria often results in triangulations with many slivers (i.e., thin and elongated triangles), which can cause several types of problems. More recently, in the field of computational geometry, higher order Delaunay triangulations (HODTs) were introduced, trying to tackle both issues at the same time-data-dependent criteria and good triangle shape-by combining data-dependent criteria with a relaxation of the Delaunay criterion. In this paper, we present the first extensive experimental study on the practical use of HODTs, as a tool to build data-dependent TINs. We present experiments with two USGS 30m digital elevation models that show that the use of HODTs can give significant improvements over the Delaunay triangulation for the criteria previously identified as most important for data-dependent triangulations, often with only a minor increase in running times. The triangulations produced have measure values comparable to those obtained with pure data-dependent approaches, without compromising the shape of the triangles, and can be computed much faster.

This paper presents several parallel algorithms for the construction of the Delaunay triangulation in E 2 and E 3 -one of the fundamental problems in computer graphics. The proposed algorithms are designed for parallel systems with shared memory and several processors. Such a hardware configuration (especially the case with two-processors) became widely spread in the last few years in the computer graphics area. Some of the proposed algorithms are easy to be implemented but not very efficient, while some of them prove opposite characteristics. Some of them are usable in E 2 only, other work in E 3 as well. The algorithms themselves were already published in computer graphics where the computer graphics criteria were highlighted. This paper concentrates on parallel and systematic point of view and gives detailed information about the parallelization of a computational geometry application to parallel and distributed computation oriented community.