A POINT-PLACEMENT STRATEGY FOR CONFORMING DELAUNAY TETRAHEDRALIZATION

2001 ◽  
Vol 11 (06) ◽  
pp. 669-682 ◽  
Author(s):  
MICHAEL MURPHY ◽  
DAVID M. MOUNT ◽  
CARL W. GABLE
Keyword(s):  
Delaunay Triangulation ◽  
Line Graph ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Complex ◽  
Straight Line ◽  
Delaunay Tetrahedralization ◽  
Planar Straight Line Graph ◽  
Set Of Points ◽  
Point Placement

A strategy is presented to find a set of points that yields a Conforming Delaunay tetrahedralization of a three-dimensional Piecewise-Linear complex (PLC). This algorithm is novel because it imposes no angle restrictions on the input PLC. In the process, an algorithm is described that computes a planar conforming Delaunay triangulation of a Planar Straight-Line Graph (PSLG) such that each triangle has a bounded circumradius, which may be of independent interest.

Finding a Covering Triangulation Whose Maximum Angle is Provably Small

1997 ◽  
Vol 07 (01n02) ◽  
pp. 5-20 ◽  
Author(s):  
Scott A. Mitchell
Keyword(s):  
Line Graph ◽  
Local Geometry ◽  
Common Boundary ◽  
Straight Line ◽  
Maximum Angle ◽  
Multiple Regions ◽  
Explicit Lower Bound ◽  
Vertex Set ◽  
Edge Covering ◽  
Planar Straight Line Graph

We consider the following problem: given a planar straight-line graph, find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem differs from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound γopt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle γ is provably close to γopt. Specifically, we show that [Formula: see text], i.e., our γ is not much closer to π than is γopt. To our knowledge, this result represents the first nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log n), where n is the number of vertices of the input. We have implemented an O(n2) time version of our algorithm.

DELAUNAY REFINEMENT ALGORITHMS FOR ESTIMATING LOCAL FEATURE SIZE IN 2D AND 3D

2011 ◽  
Vol 21 (05) ◽  
pp. 507-543
Author(s):  
ALEXANDER RAND ◽  
NOEL WALKINGTON
Keyword(s):  
Delaunay Triangulation ◽  
Mesh Generation ◽  
Piecewise Linear ◽  
Local Information ◽  
Local Feature ◽  
Delaunay Refinement ◽  
Linear Complex ◽  
Feature Size ◽  
2D And 3D ◽  
Local Feature Size

We present Delaunay refinement algorithms for estimating local feature size on the input vertices of a 2D piecewise linear complex and on the input vertices and segments of a 3D piecewise linear complex. These algorithms are designed to eliminate the need for a local feature size oracle during quality mesh generation of domains containing acute input angles. In keeping with Ruppert's algorithm, encroachment in these algorithms can be determined through only local information in the current Delaunay triangulation. The algorithms are practical to implement and several examples are given.

scholarly journals ANY FIVE POINTS LIE IN A 3-SPACE

2020 ◽  
Author(s):  
Zh. Nikoghosyan ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Entire Space ◽  
Unique Line ◽  
Straight Line ◽  
Set Of Points ◽  
Three Dimensional Space

In axiomatic formulations, every two points lie in a (straight) line, every three points lie in a plane and every four points lie in a three-dimensional space (3-space). In this paper we show that every five points lie in a 3-space as well, implying that every set of points lie in a 3-space. In other words, the 3-space occupies the entire space. The proof is based on the following four axioms: 1) every two distinct points define a unique line, 2) every three distinct points, not lying on the line, define a unique plane, 3) if 𝐴 and 𝐵 are two distinct points in a 3-space, then the line defined by the points 𝐴, 𝐵, entirely lies in this 3-space, 4) if 𝐹1, 𝐹2, 𝐹3 are three distinct points in a 3-space, not lying in a line, then the plane defined by the points 𝐹1, 𝐹2, 𝐹3, lies entirely in this 3-space.

scholarly journals Convex Polygons in Geometric Triangulations

2017 ◽  
Vol 26 (5) ◽  
pp. 641-659 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
CSABA D. TÓTH
Keyword(s):  
Lower Bound ◽  
Line Graph ◽  
Convex Polygons ◽  
Straight Line ◽  
Planar Straight Line Graph

We show that the maximum number of convex polygons in a triangulation ofnpoints in the plane isO(1.5029n). This improves an earlier bound ofO(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graphGwithnvertices, we also show how to compute efficiently the number of convex polygons inG.

THREE-DIMENSIONAL SHAPES OF A FINITE SET OF POINTS

2003 ◽  
Vol 17 (02) ◽  
pp. 301-318 ◽  
Author(s):  
MAHMOUD MELKEMI
Keyword(s):  
Voronoi Diagram ◽  
Delaunay Triangulation ◽  
Efficient Algorithm ◽  
Point Cloud ◽  
Three Dimensional ◽  
Levels Of Detail ◽  
Finite Set ◽  
Set Of Points ◽  
Different Levels ◽  
Different Parts

The three-dimensional [Formula: see text]-shape is based on a mathematical formalism which determines exact relationships between points and shapes. It reconstructs surface and volume and detects 3D dot patterns for a given point cloud. [Formula: see text]-shape of a set of points is a sub-complex of Delaunay triangulation of this set. It generates a family of shapes according to the selected [Formula: see text] (a set of points). A method to compute the positions of the points of [Formula: see text] is proposed. These points are selected from the vertices of Voronoi diagram by analyzing the form of the polytopes; their elongation. This method allows the [Formula: see text]-shape to reflect different levels of detail in different parts of space. An efficient algorithm computing the three-dimensional [Formula: see text]-shape is presented, the [Formula: see text]-shape of a set of points is derived from the Delaunay triangulation of the same set. The speed of the algorithm is determined by the speed of the algorithm computing the Delaunay triangulation.

WHEN AND WHY DELAUNAY REFINEMENT ALGORITHMS WORK

2005 ◽  
Vol 15 (01) ◽  
pp. 25-54 ◽  
Author(s):  
GARY L. MILLER ◽  
STEVEN E. PAV ◽  
NOEL J. WALKINGTON
Keyword(s):  
Line Graph ◽  
Delaunay Refinement ◽  
Straight Line ◽  
Minimum Angle ◽  
Planar Straight Line Graph ◽  
Delaunay Mesh ◽  
Concentric Shell ◽  
Ruppert's Algorithm

An "adaptive" variant of Ruppert's Algorithm for producing quality triangular planar meshes is introduced. The algorithm terminates for arbitrary Planar Straight Line Graph (PSLG) input. The algorithm outputs a Delaunay mesh where no triangle has minimum angle smaller than about 26.45° except "across" from small angles of the input. No angle of the output mesh is smaller than arctan [(sin θ*)/(2-cos θ*)] where θ* is the minimum input angle. Moreover no angle of the mesh is larger than about 137°, independent of small input angles. The adaptive variant is unnecessary when θ* is larger than 36.53°, and thus Ruppert's Algorithm (with concentric shell splitting) can accept input with minimum angle as small as 36.53°. An argument is made for why Ruppert's Algorithm can terminate when the minimum output angle is as large as 30°.

Viewpoint-based shape recovery from multiple views

2000 ◽  
Vol 14 (5) ◽  
pp. 359-372
Author(s):  
ALI AKGUNDUZ ◽  
DAN ZETU ◽  
PAT BANERJEE
Keyword(s):  
Convex Hull ◽  
Surface Reconstruction ◽  
Delaunay Triangulation ◽  
Shape Recovery ◽  
Three Dimensional ◽  
Image Sequence ◽  
Multiple Views ◽  
Recovery Algorithm ◽  
Additional Information ◽  
Set Of Points

This paper describes an algorithm for surface reconstruction from a set of scattered three-dimensional points extracted from an image sequence. In this process, additional information (such as location of the viewpoints and the points visible from a particular viewpoint) is available and can be exploited for accurately recovering the shape of the objects portrayed in the images. Initially, the set of points is subjected to a Delaunay triangulation that fills the convex hull of the set of points with disjoint tetrahedra. The key idea of the shape recovery algorithm is to eliminate triangles that obstruct the visibility of points from certain viewpoints, whose locations are known from the image acquisition process. The major contribution of this paper is that we have been able to design an algorithm for surface reconstruction that handles a wide variety of shapes, as opposed to currently existing techniques.

A Constrained Delaunay Triangulation Algorithm Based on Incremental Points

2011 ◽  
Vol 90-93 ◽  
pp. 3277-3282 ◽  
Author(s):  
Bai Chao Wu ◽  
Ai Ping Tang ◽  
Lian Fa Wang
Keyword(s):  
Information System ◽  
Data Structures ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Three Dimensional ◽  
Geographical Information ◽  
Two Dimensional ◽  
Data Set ◽  
Constrained Delaunay Triangulation

The foundation ofdelaunay triangulationandconstrained delaunay triangulationis the basis of three dimensional geographical information system which is one of hot issues of the contemporary era; moreover it is widely applied in finite element methods, terrain modeling and object reconstruction, euclidean minimum spanning tree and other applications. An algorithm for generatingconstrained delaunay triangulationin two dimensional planes is presented. The algorithm permits constrained edges and polygons (possibly with holes) to be specified in the triangulations, and describes some data structures related to constrained edges and polygons. In order to maintain the delaunay criterion largely，some new incremental points are added onto the constrained ones. After the data set is preprocessed, the foundation ofconstrained delaunay triangulationis showed as follows: firstly, the constrained edges and polygons generate initial triangulations，then the remained points completes the triangulation . Some pseudo-codes involved in the algorithm are provided. Finally, some conclusions and further studies are given.

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