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Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.


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Journal ArticleDOI
TL;DR: It is argued that an aspect ratio is good if the Delaunay triangulation of the scatter plot at this aspect ratio has some nice geometric property, e.g., a large minimum angle or a small total edge length.
Abstract: Scatter plots are diagrams that visualize two-dimensional data as sets of points in the plane They allow users to detect correlations and clusters in the data Whether or not a user can accomplish these tasks highly depends on the aspect ratio selected for the plot, ie, the ratio between the horizontal and the vertical extent of the diagram We argue that an aspect ratio is good if the Delaunay triangulation of the scatter plot at this aspect ratio has some nice geometric property, eg, a large minimum angle or a small total edge length More precisely, we consider the following optimization problem Given a set Q of points in the plane, find a scale factor s such that scaling the x-coordinates of the points in Q by s and the y-coordinates by 1=s yields a point set P(s) that optimizes a property of the Delaunay triangulation of P(s), over all choices of s We present an algorithm that solves this problem efficiently and demonstrate its usefulness on real-world instances Moreover, we discuss an empirical test in which we asked 64 participants to choose the aspect ratios of 18 scatter plots We tested six different quality measures that our algorithm can optimize In conclusion, minimizing the total edge length and minimizing what we call the 'uncompactness' of the triangles of the Delaunay triangulation yielded the aspect ratios that were most similar to those chosen by the participants in the test

30 citations

Journal ArticleDOI
Tamal K. Dey1, Xiaoyin Ge1, Qichao Que1, Issam Safa1, Lei Wang1, Yusu Wang1 
TL;DR: This paper allows the presence of various singularities by requiring that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect.
Abstract: Reconstructing a surface mesh from a set of discrete point samples is a fundamental problem in geometric modeling. It becomes challenging in presence of ‘singularities’ such as boundaries, sharp features, and non-manifolds. A few of the current research in reconstruction have addressed handling some of these singularities, but a unified approach to handle them all is missing. In this paper we allow the presence of various singularities by requiring that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect. Our algorithm first identifies and reconstructs the features where singularities occur. Next, it reconstructs the surface patches containing these feature curves. The identification and reconstruction of feature curves are achieved by a novel combination of the Gaussian weighted graph Laplacian and the Reeb graphs. The global reconstruction is achieved by a method akin to the well known Cocone reconstruction, but with weighted Delaunay triangulation that allows protecting the feature samples with balls. We provide various experimental results to demonstrate the effectiveness of our feature-preserving singular surface reconstruction algorithm. © 2012 Wiley Periodicals, Inc.

30 citations

Proceedings ArticleDOI
13 Jun 2010
TL;DR: The notion of a stable Delaunay graph (SDG in short) is introduced, a dynamic subgraph of the Delauny triangulation that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunays.
Abstract: The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℜ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter ± > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least ±. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O*(n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/± and polylogarithmic in n. The first structure is simpler but the dependency on 1/± in its performance is higher.

30 citations

Journal ArticleDOI
TL;DR: A method of ray tracing for free-form optical surfaces is developed based on Delaunay triangulation of the discrete data of the surface and is related to finite-element modeling.
Abstract: A method of ray tracing for free-form optical surfaces has been developed. The ray tracing through such surfaces is based on Delaunay triangulation of the discrete data of the surface and is related to finite-element modeling. Some numerical examples of applications to analytical, noisy, and experimental free-form surfaces (in particular, a corneal topography map) are presented. Ray-tracing results (i.e., spot diagram root-mean-square error) with the new method are in agreement with those obtained using a modal fitting of the surface, for sampling densities higher than 40 x 40 elements. The method competes in flexibility, simplicity, and computing times with standard methods for surface fitting and ray tracing.

30 citations

Book ChapterDOI
16 Jun 1997
TL;DR: This paper presents a new method for progressively previewing a ray-traced image while it is being computed, which constructs and incrementally updates a constrained Delaunay triangulation of the image plane.
Abstract: This paper presents a new method for progressively previewing a ray-traced image while it is being computed. Our method constructs and incrementally updates a constrained Delaunay triangulation of the image plane. The points in the triangulation correspond to all of the image samples that have been computed by the ray tracer, and the constraint edges correspond to various important discontinuity edges in the image. The triangulation is displayed using hardware Gouraud shading, yielding a piecewise-linear approximation to the final image. Texture mapped surfaces, as well as other regions in the image that are not well approximated by linear interpolation, are handled with the aid of hardware texture mapping.

30 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202393
2022203
2021130
2020185
2019204
2018223