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

Discrete physics using metrized chains

Over the last fifty years, there have been numerous efforts to develop comprehensive discrete formulations of geometry and physics from first principles: from Whitney's geometric integration theory [33] to Harrison's theory of chainlets [16], including Regge calculus in general relativity [26, 34], Tonti's work on the mathematical structure of physical theories [30] and their discrete formulation [31], plus multifarious researches into so-called mimetic discretization methods [28], discrete exterior calculus [11, 12], and discrete differential geometry [2, 10]. All these approaches strive to tell apart the different mathematical structures---topological, differentiable, metrical---underpinning a physical theory, in order to make the relationships between them more transparent. While each component is reasonably well understood, computationally effective connections between them are not yet well established, leading to difficulties in combining and progressively refining geometric models and physics-based simulations. This paper proposes such a connection by introducing the concept of metrized chains, meant to establish a discrete metric structure on top of a discrete measure-theoretic structure embodied in the underlying notion of measured (real-valued) chains. These, in turn, are defined on a cell complex, a finite approximation to a manifold which abstracts only its topological properties. The algebraic-topological approach to circuit design and network analysis first proposed by Branin [7] was later extensively applied by Tonti to the study of the mathematical structure of physical theories [30]. (Co-)chains subsequently entered the field of physical modeling [4, 18, 24, 25, 31, 37], and were related to commonly-used discretization methods such as finite elements, finite differences, and finite volumes [1, 8, 21, 22]. Our modus operandi is characterized by the pivotal role we accord to the construction of a physically based inner product between chains. This leads us to criticize the emphasis placed on the choice of an appropriate dual mesh: in our opinion, the "good" dual mesh is but a red herring, in general