Kinetic and dynamic Delaunay tetrahedralizations in three dimensions

Abstract

We describe the implementation of algorithms to construct and maintain threedimensional dynamic Delaunay triangulations with kinetic vertices using a threesimplex data structure. The code is capable of constructing the geometric dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points is triangulated. Time evolution of the triangulation is not only governed by kinetic vertices but also by a changing number of vertices. We use three-dimensional simplex flip algorithms, a stochastic visibility walk algorithm for point location and in addition, we propose a new simple method of deleting vertices from an existing three-dimensional Delaunay triangulation while maintaining the Delaunay property. The dual Dirichlet tessellation can be used to solve differential equations on an irregular grid, to define partitions in cell tissue simulations, for collision detection etc.Key words: Delaunay triangulation, Dirichlet, Voronoi, flips, point location, vertex deletion, kinetic algorithm, dynamic algorithmIn nearly all aspects of science nowadays simulations of discrete objects underlying different interactions play a very important role. Such an interaction for example could be mediated by colliding grains of sand in an hourglass or -more abstract -the neighborhood question of influence regions. One general method to represent possible two-body interactions within a system of N objects is given by a network which can be described by an N × N adjacency matrix ν, with its matrix elements ν ij = ν ji (undirected graph) representing the interaction between the objects i and j. However, for most realistic systems the graph defined this way is not practical if one remembers that the typical size of a system of atoms in chemistry can be O (10 23 ), the human body consists of O (10 18 ) cells and even simple systems such as a grain-filled hourglass

Modeling emergent tissue organization involving high-speed migrating cells in a flow equilibrium

“…The tissue formation problem of migrating cells is simulated using an agent-based model on top of a regular triangulation [15,16,23,24]. The regular triangulation is used to provide the neighborhood topology for the cells that allows for a continuous representation of cell positions and sizes in contrast to grid-based methods.…”

Data Analysis in Cosmology

“…The work by Meyer-Hermann and collaborators (Schaller & Meyer-Hermann, 2004;Beyer et al, 2005;Beyer & Meyer-Hermann, 2006) may prove to represent a watershed in having managed to define a parallel code for the kinetic and dynamic calculation of Delaunay grids. Once the Delaunay grid of the initial point configuration has been computed, subsequent timesteps involve a continuous dynamical upgrade via local Delaunay simplex upgrades as points move around and switch the identity of their natural neighbours.…”

Towards Sub-cellular Modeling with Delaunay Triangulation

“…For a triangulation to be considered a Delaunay triangulation, all of its simplices must satisfy the empty-circumsphere-criterion, i.e., no vertex of the triangulation may lie inside the circumsphere of the triangulation simplices. Therefore, a Delaunay triangulation is uniquely defined if the points in P are in extended general position, i.e., no two points are identical, no three points lie on a common line, no four points lie on the same plane and no five points lie on a common sphere (for the three-dimensional case discussed here) [19,21,22,23].…”

“…The model described here is an extension on a framework developed in the last years [19,20]. The program generates kinetic (moving vertices) and dynamic (changing number of vertices) Delaunay triangulations for a set of agents, being particularly useful for the simulation of evolving biological systems, where the number of agents varies and at every time step the neighborhood relations must be adjusted.…”

“…The underlying algorithms to calculate Delaunay triangulations used in this work have been published before [19,20,24] and are available upon request. The implementation is a fully kinetic and dynamic three-dimensional Delaunay triangulation of a given set of particles, and supports dynamic insertion and deletion of vertices, being therefore specially suitable for defining interaction relations between particles in systems where the number of particles may vary.…”