Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Simple and Robust Boolean Operations for Triangulated Surfaces

Boolean operations of geometric models is an essential issue in computational geometry. In this paper, we develop a simple and robust approach to perform Boolean operations on closed and open triangulated surfaces. Our method mainly has two stages: (1) We firstly find out candidate intersected-triangles pairs based on Octree and then compute the inter-section lines for all pairs of triangles with parallel algorithm; (2) We form closed or open intersection-loops, sub-surfaces and sub-blocks quite robustly only according to the cleared and updated topology of meshes while without coordinate computations for geometric enti-ties. A novel technique instead of inside/outside classification is also proposed to distinguish the resulting union, subtraction and intersection. Several examples have been given to illus-trate the effectiveness of our approach.Comment: Novel method for determining Union, Subtraction and Intersectio

Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain. A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task. In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed B\'ezier trivariates, at all its internal knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International Conference on Geometric Modeling and Processing (GMP 2019

Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)

The alternating sum of volumes (ASV) decomposition is a widely used technique for converting a B-Rep into a CSG model. The obtained CSG tree has convex primitives at its leaf nodes, while the contents of its internal nodes alternate between the set union and difference operators. This work first shows that the obtained CSG tree T can also be expressed as the regularized Exclusive-OR operation among all the convex primitives at the leaf nodes of T, regardless the structure and internal nodes of T. This is an important result in the case in which EVM represented orthogonal polyhedra are used because in this model the Exclusive-OR operation runs much faster than set union and difference operations. Therefore this work applies this result to EVM represented orthogonal polyhedra. It also presents experimental results that corroborate the theoretical results and includes some practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version

Robust solid modeling by avoiding redundancy for manifold objects in boundary representation

Journal ArticleThis paper describes a new approach to the robustness problem in solid modeling. We identify as t h e main cause of t h e lack of robustness that interdependent topological relations are derived from approximate data. Disregarding the interdependencies very likely violates basic properties, such as reflexivity, and transitivity, resulting in invalid data representations, such as dangling edges, missing faces, etc. We show that the boundary of manifold objects can be represented without redundant relations which avoids inconsistencies. An algorithm for regularized set operations for manifold solids which is based on the principle of avoiding and eliminating redundancy is described. This algorithm has been implemented for objects bounded by planar and natural quadric surfaces; it handles coincidence and incidence cases between surfaces and curves robustly