Preferred directions for resolving the non-uniqueness of Delaunay triangulations

AbstractThis note proposes a simple rule to determine a unique triangulation among all Delaunay triangulations of a planar point set, based on two preferred directions. We show that the triangulation can be generated by extending Lawson's edge-swapping algorithm and that point deletion is a local procedure. The rule can be implemented exactly when the points have integer coordinates and can be used to improve image compression methods

Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees

pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach

Qualitative Symbolic Perturbation: a new geometry-based perturbation framework

In a classical Symbolic Perturbation scheme,degeneracies are handled by substituting some polynomials in$\varepsilon$ to the input of a predicate. Instead of a singleperturbation, we propose to use a sequence of (simpler)perturbations. Moreover, we look at their effects geometricallyinstead of algebraically; this allows us to tackle cases that werenot tractable with the classical algebraic approach.Avec les méthodes de perturbations symboliques classiques,les dégénérescences sont résolues en substituant certains polynômes en $\varepsilon$ aux entrées du prédicat.Au lieu d'une seule perturbation compliquée, nous proposons d'utiliser unesuite de perturbation plus simple. Et nous regardons les effets deces perturbations géométriquement plutôt qu'algébriquementce qui permet de traiter des cas inatteignables par les méthodesalgébriques classiques

Topology verification for isosurface extraction

Journal ArticleThe broad goals of verifiable visualization rely on correct algorithmic implementations. We extend a framework for verification of isosurfacing implementations to check topological properties. Specifically, we use stratified Morse theory and digital topology to design algorithms which verify topological invariants. Our extended framework reveals unexpected behavior and coding mistakes in popular publicly available isosurface codes