# Triangulation - Explore the Science & Experts | ideXlab

## Triangulation

The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform

### Rodrigo I. Silveira – One of the best experts on this subject based on the ideXlab platform.

• ##### Implementing data-dependent Triangulations with higher order delaunay Triangulations
ISPRS international journal of geo-information, 2017
Co-Authors: Natalia Rodríguez, Rodrigo I. Silveira
Abstract:

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.

• ##### SIGSPATIAL/GIS – Implementing data-dependent Triangulations with higher order Delaunay Triangulations
Proceedings of the 24th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems – GIS '16, 2016
Co-Authors: Natalia Rodríguez, Rodrigo I. Silveira
Abstract:

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. It has long been recognized that sometimes it may be beneficial to use other, non-Delaunay, criteria to build TINs. Data-dependent Triangulations were introduced decades ago to address this. However, they are rarely used in practice, mostly because the optimization of data- dependent criteria often results in Triangulations with many thin and elongated triangles. 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. Nevertheless, most previous studies about them have been limited to theoretical aspects. In this work 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 terrains that show that HODTs can give significant improvements over the Delaunay Triangulation for the criteria identified as most important for data-dependent Triangulations. The resulting Triangulations have data-dependent values comparable to those obtained with pure data-dependent approaches, without compromising the shape of the triangles, and are faster to compute.

• ##### On the Number of Higher Order Delaunay Triangulations
Theoretical Computer Science, 2011
Co-Authors: Dieter Mitsche, Maria Saumell, Rodrigo I. Silveira
Abstract:

Higher order Delaunay Triangulations are a generalization of the Delaunay Triangulation which provides a class of well-shaped Triangulations, over which extra criteria can be optimized. A Triangulation is order-k Delaunay if the circumcircle of each triangle of the Triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay Triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay Triangulation, even for large orders, whereas for first order Delaunay Triangulations, the maximum number is 2^{n−3}. Next we show that uniformly distributed points have an expected number of at least 2^{\rho_1 n(1+o(1))} first order Delaunay Triangulations, where \rho_1 is an analytically defined constant (\rho_1 \sim 0.525785), and for k>1, the expected number of order-k Delaunay Triangulations (which are not order-i for any i

### Mouaaz Nahas – One of the best experts on this subject based on the ideXlab platform.

• ##### application layer multicast with delaunay Triangulations
Global Communications Conference, 2001
Co-Authors: Jorg Liebeherr, Mouaaz Nahas
Abstract:

Recently, application-layer multicast has emerged as an attempt to support group applications without the need for a network-layer multicast protocol, such as IP multicast. In application-layer multicast, applications arrange themselves as a logical overlay network and transfer data within the overlay network. In this paper, Delaunay Triangulations are investigated as an overlay network topology for application-layer multicast. An advantage of Delaunay Triangulations is that each application can locally derive next-hop routing information without the need for a routing protocol in the overlay. A disadvantage of a Delaunay Triangulation as an overlay topology is that the mapping of the overlay to the network-layer infrastructure may be suboptimal. It is shown that this disadvantage can be partially addressed with a hierarchical organization of Delaunay Triangulations. Using network topology generators, the Delaunay Triangulation is compared to other proposed overlay topologies for application-layer multicast.

• ##### GLOBECOM – Application-layer multicast with Delaunay Triangulations
GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270), 2001
Co-Authors: Jorg Liebeherr, Mouaaz Nahas
Abstract:

Recently, application-layer multicast has emerged as an attempt to support group applications without the need for a network-layer multicast protocol, such as IP multicast. In application-layer multicast, applications arrange themselves as a logical overlay network and transfer data within the overlay network. In this paper, Delaunay Triangulations are investigated as an overlay network topology for application-layer multicast. An advantage of Delaunay Triangulations is that each application can locally derive next-hop routing information without the need for a routing protocol in the overlay. A disadvantage of a Delaunay Triangulation as an overlay topology is that the mapping of the overlay to the network-layer infrastructure may be suboptimal. It is shown that this disadvantage can be partially addressed with a hierarchical organization of Delaunay Triangulations. Using network topology generators, the Delaunay Triangulation is compared to other proposed overlay topologies for application-layer multicast.

### Peter Oswald – One of the best experts on this subject based on the ideXlab platform.

• ##### Nonconforming P1 elements on distorted Triangulations: Lower bounds for the discrete energy norm error
Applications of Mathematics, 2017
Co-Authors: Peter Oswald
Abstract:

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted Triangulations. E.g., optimal-order discrete H ^1 norm best approximation error estimates for H ^2 functions hold for arbitrary Triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the Triangulation. We demonstrate on an example of a special family of distorted Triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of Triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.

• ##### nonconforming p1 elements on distorted Triangulations lower bounds for the discrete energy norm error
arXiv: Numerical Analysis, 2016
Co-Authors: Peter Oswald
Abstract:

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted Triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary Triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the Triangulation. We demonstrate on the example of a special family of distorted Triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of Triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily slow speed. The results complement analogous findings for conforming P1 elements.

### Camilo Sarmiento – One of the best experts on this subject based on the ideXlab platform.

• ##### Dyck path Triangulations and extendability (extended abstract)
Discrete Mathematics and Theoretical Computer Science, 2015
Co-Authors: Cesar Ceballos, Arnau Padrol, Camilo Sarmiento
Abstract:

We introduce the Dyck path Triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this Triangulation are given by Dyck paths, and its construction naturally generalizes to produce Triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path Triangulation is motivated by extendability problems of partial Triangulations of products of two simplices. We show that whenever$m\geq k>n$, any Triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique Triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.

• ##### Dyck path Triangulations and extendability
Journal of Combinatorial Theory Series A, 2015
Co-Authors: Cesar Ceballos, Arnau Padrol, Camilo Sarmiento
Abstract:

We introduce the Dyck path Triangulation of the cartesian product of two simplices Δ n – 1 i? Δ n – 1 . The maximal simplices of this Triangulation are given by Dyck paths, and the construction naturally generalizes to certain rational Dyck paths. Our study of the Dyck path Triangulation is motivated by extendability problems of partial Triangulations of products of two simplices. We show that whenever m ? k n , any Triangulation of the product of the k-skeleton of Δ m – 1 with Δ n – 1 extends to a unique Triangulation of Δ m – 1 i? Δ n – 1 . Moreover, using the Dyck path Triangulation, we prove that the bound k n is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids that are analogous to classical results in oriented matroid theory.

• ##### Dyck path Triangulations and extendability
arXiv: Combinatorics, 2014
Co-Authors: Cesar Ceballos, Arnau Padrol, Camilo Sarmiento
Abstract:

We introduce the Dyck path Triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this Triangulation are given by Dyck paths, and its construction naturally generalizes to produce Triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path Triangulation is motivated by extendability problems of partial Triangulations of products of two simplices. We show that whenever $m\geq k>n$, any Triangulation of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique Triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interesting interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.

### Jorg Liebeherr – One of the best experts on this subject based on the ideXlab platform.

• ##### application layer multicast with delaunay Triangulations
Global Communications Conference, 2001
Co-Authors: Jorg Liebeherr, Mouaaz Nahas
Abstract:

Recently, application-layer multicast has emerged as an attempt to support group applications without the need for a network-layer multicast protocol, such as IP multicast. In application-layer multicast, applications arrange themselves as a logical overlay network and transfer data within the overlay network. In this paper, Delaunay Triangulations are investigated as an overlay network topology for application-layer multicast. An advantage of Delaunay Triangulations is that each application can locally derive next-hop routing information without the need for a routing protocol in the overlay. A disadvantage of a Delaunay Triangulation as an overlay topology is that the mapping of the overlay to the network-layer infrastructure may be suboptimal. It is shown that this disadvantage can be partially addressed with a hierarchical organization of Delaunay Triangulations. Using network topology generators, the Delaunay Triangulation is compared to other proposed overlay topologies for application-layer multicast.

• ##### GLOBECOM – Application-layer multicast with Delaunay Triangulations
GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270), 2001
Co-Authors: Jorg Liebeherr, Mouaaz Nahas
Abstract:

Recently, application-layer multicast has emerged as an attempt to support group applications without the need for a network-layer multicast protocol, such as IP multicast. In application-layer multicast, applications arrange themselves as a logical overlay network and transfer data within the overlay network. In this paper, Delaunay Triangulations are investigated as an overlay network topology for application-layer multicast. An advantage of Delaunay Triangulations is that each application can locally derive next-hop routing information without the need for a routing protocol in the overlay. A disadvantage of a Delaunay Triangulation as an overlay topology is that the mapping of the overlay to the network-layer infrastructure may be suboptimal. It is shown that this disadvantage can be partially addressed with a hierarchical organization of Delaunay Triangulations. Using network topology generators, the Delaunay Triangulation is compared to other proposed overlay topologies for application-layer multicast.