scholarly journals Constrained boundary recovery for three dimensional Delaunay triangulations

2004 ◽  
Vol 61 (9) ◽  
pp. 1471-1500 ◽  
Author(s):  
Qiang Du ◽  
Desheng Wang
Keyword(s):  
Three Dimensional ◽  
Delaunay Triangulations

scholarly journals Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 356 ◽  
Author(s):  
Jose Diaz-Severiano ◽  
Valentin Gomez-Jauregui ◽  
Cristina Manchado ◽  
Cesar Otero
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Inversive Plane ◽  
Delaunay Triangulations ◽  
Möbius Transformations ◽  
Regular Polyhedra ◽  
Complex Design ◽  
Elliptic Geometry ◽  
Mobius Transformations

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

A Robust Implementation for Three-Dimensional Delaunay Triangulations

1998 ◽  
Vol 08 (02) ◽  
pp. 255-276 ◽  
Author(s):  
Ernst P. Mücke
Keyword(s):  
Time Complexity ◽  
Source Code ◽  
Three Dimensional ◽  
The Internet ◽  
Perturbation Scheme ◽  
Worst Case ◽  
Delaunay Triangulations ◽  
Exact Arithmetic ◽  
Robust Implementation ◽  
Geometric Primitives

This paper presents an implementation for Delaunay triangulations of three-dimensional point sets. The code uses a variant of the randomized incremental flip algorithm and employs symbolic perturbation to achieve robustness. The algorithm's theoretical time complexity is quadratic in n, the number of input points, and this is optimal in the worst case. However, empirical running times are proportional to the number of triangles in the final triangulation, which is typically linear in n. Even though the symbolic perturbation scheme relies on exact arithmetic, the resulting code is efficient in practice. This is due to a careful implementation of the geometric primitives and the arithmetic module. The source code is available on the Internet.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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