Sparse implicitization by interpolation: Geometric computations using matrix representations

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation

Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton-Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision surface, and is convertible into a collection of B\'ezier patches. First, a hierarchical bounding volume tree is used to efficiently identify the B\'ezier patches with a convex-hull intersecting the convex-hull of a given curve segment. For ease of implementation convex-hulls are approximated with k-dops (discrete orientation polytopes). Subsequently, the intersections of the identified B\'ezier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular value decompositions (SVDs). As an application of the developed interrogation technique the isogeometric design and analysis of lattice-skin structures is investigated. The skin is a spline surface that is usually created in a computer-aided design (CAD) system and the periodic lattice to be fitted consists of unit cells, each containing a small number of struts. The lattice-skin structure is generated by projecting selected lattice nodes onto the surface after determining the intersection of unit cell edges with the surface. For mechanical analysis, the skin is modelled as a Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss. The two types of structures are coupled with a standard Lagrange multiplier approach

Intersection entre courbes et surfaces rationnelles au moyen des représentations implicites matricielles

National audienceDans cet article, on introduit une nouvelle représentation implicite des courbes et des surfaces paramétrées rationelles, représentation qui consiste pour l'essentiel à les caractériser par la chute de rang d'une matrice plutôt que par l'annulation simultanée d'une ou plusieurs équations polynomiales. On montre comment ces représentations implicites, que l'on qualifiera de matricielles, établissent un pont entre la géométrie et l'algèbre linéaire, pont qui permet de livrer des problèmes géométriques à des algorithmes classiques et éprouvés d'algèbre linéaire, ouvrant ainsi la possibilité d'un traitement numérique plus robuste. La contribution de cette approche est discutée et illustrée sur des problèmes importants de la modélisation géométrique tels que la localisation (appartenance d'un point à un objet), le calcul d'intersection de deux objets, ou bien encore la détection d'un lieu singulier

A Line/Trimmed NURBS Surface Intersection Algorithm Using Matrix Representations

International audienceWe contribute a reliable line/surface intersection method for trimmed NURBS surfaces, based on a novel matrix-based implicit representation and numerical methods in linear algebra such as singular value decomposition and the computation of generalized eigenvalues and eigenvectors. A careful treatment of degenerate cases makes our approach robust to intersection points with multiple pre-images. We then apply our intersection algorithm to mesh NURBS surfaces through Delaunay refinement. We demonstrate the added value of our approach in terms of accuracy and treatment of degenerate cases, by providing comparisons with other intersection approaches as well as a variety of meshing experiments