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ISPRS Int. J. Geo-Inf. 2017, 6(12), 390; https://doi.org/10.3390/ijgi6120390

Implementing Data-Dependent Triangulations with Higher Order Delaunay Triangulations

1
Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires C1428EGA, Argentina
2
Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
This paper is an extended version of our paper published in 24th SIGSPATIAL International Conference on Advances in Geographic Information Systems (SIGSPATIAL 2016).
*
Author to whom correspondence should be addressed.
Received: 18 September 2017 / Revised: 23 November 2017 / Accepted: 26 November 2017 / Published: 1 December 2017
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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. View Full-Text
Keywords: data-dependent triangulation; Delaunay triangulation; higher order Delaunay triangulation; triangulated irregular network; digital terrain model data-dependent triangulation; Delaunay triangulation; higher order Delaunay triangulation; triangulated irregular network; digital terrain model
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Rodríguez, N.; Silveira, R.I. Implementing Data-Dependent Triangulations with Higher Order Delaunay Triangulations. ISPRS Int. J. Geo-Inf. 2017, 6, 390.

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