Topic
Point set triangulation
About: Point set triangulation is a research topic. Over the lifetime, 566 publications have been published within this topic receiving 12761 citations.
Papers published on a yearly basis
Papers
TL;DR: If the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation and the algorithm takes expected time at mostO(nlogn+n[d/2]).
Abstract: A set ofn weighted points in general position in źd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.
442 citations
TL;DR: An algorithm is developed to iteratively remove triangles from the triangulation using an underlying parametric or implicit surface representation and an order is introduced on the set of triangles by considering curvature at their vertices.
311 citations
TL;DR: In this article, it was shown that the decision version of this problem is NP-hard, using a reduction from PLANAR 1-IN-3-SAT, and the correct working of the gadgets was established with computer assistance, using dynamic programming on polygonal faces, as well as the β-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.
Abstract: A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard, using a reduction from PLANAR 1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the β-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.
203 citations
TL;DR: A new hierarchical triangle-based model for representing surfaces over sampled data, based on the subdivision of the surface domain into nested triangulations, is proposed, which allows compression of spatial data and representation of a surface at successively finer degrees of resolution.
Abstract: A new hierarchical triangle-based model for representing surfaces over sampled data is proposed, which is based on the subdivision of the surface domain into nested triangulations, called a hierarchical triangulation (HT). The model allows compression of spatial data and representation of a surface at successively finer degrees of resolution. An HT is a collection of triangulations organized in a tree, where each node, except for the root, is a triangulation refining a face belonging to its parent in the hierarchy. We present a topological model for representing an HT, and algorithms for its construction and for the extraction of a triangulation at a given degree of resolution. The surface model, called a hierarchical triangulated surface (HTS) is obtained by associating data values with the vertices of triangles, and by defining suitable functions that describe the surface over each triangular patch. We consider an application of a piecewise-linear version of the HTS to interpolate topographical data, and we describe a specialized version of the construction algorithm that builds an HTS for a terrain starting from a high-resolution rectangular grid of sampled data. Finally, we present an algorithm for extracting representations of terrain at variable resolution over the domain.
192 citations
TL;DR: A number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplical tessellation of the convex hull of an arbitrary finite set of points in n-space are established.
191 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 2 |
| 2024 | 1 |
| 2023 | 3 |
| 2022 | 4 |
| 2021 | 1 |
| 2020 | 3 |