Planar hexagonal meshing for architecture

published_or_final_versio

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems

Compact data structures for triangulations

International audienceThe main problem consists in designing space-efficient data structures allowing to represent the connectivity of triangle meshes while supporting fast navigation and local updates

Polygonization of Multi-Component Non-Manifold Implicit Surfaces through A Symbolic-Numerical Continuation Algorithm

In computer graphics, most algorithms for sampling implicit surfaces use a 2-points numerical method. If the surface-describing function evaluates positive at the first point and negative at the second one, we can say that the surface is located somewhere between them. Surfaces detected this way are called sign-variant implicit surfaces. However, 2-points numerical methods may fail to detect and sample the surface because the functions of many implicit surfaces evaluate either positive or negative everywhere around them. These surfaces are here called sign-invariant implicit surfaces. In this paper, instead of using a 2-points numerical method, we use a 1-point numerical method to guarantee that our algorithm detects and samples both sign-variant and sign-invariant surface components or branches correctly. This algorithm follows a continuation approach to tessellate implicit surfaces, so that it applies symbolic factorization to decompose the function expression into symbolic components, sampling then each symbolic function component separately. This ensures that our algorithm detects, samples, and triangulates most components of implicit surfaces

A physics-based adaptive point distribution method for computational domain discretization

Two algorithms are presented which together generate well-spaced point distributions applied to curves, surfaces, and the volume of a computational domain. The first is a force equilibrium simulation which applies a simplified direct solution of the equations of motion at each node. Inter-nodal pair forces are computed based on the desired spacing between nodes and summed to provide a net force on each node. The nodes are allowed to travel a restricted distance with each locally distinct time step. The motion of the point distribution is stabilized by applying friction to each node from its neighboring nodes as well as globally restricting the time step size over the series of iterations. Second, an algorithm for node population adaptation is presented which deletes nodes or inserts new nodes depending on how well the local concentration of nodes matches a desired local spacing prescription, or spacing field. Experimental results are provided which demonstrate the ability of these algorithms to generate smooth distributions of points matching various spacing field function definitions