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

Local congruence of chain complexes

The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of {\epsilon}-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [\Delta_0] and [\Delta_1], used as matrix accumulators of the local coboundary chains, to the global matrices [\delta_0] and [\delta_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2].Comment: to submi

Progressive Conversion from B-rep to BSP for Streaming Geometric Modeling

We introduce a novel progressive approach to generate a Binary Space Partition (BSP ) and convex cell decomposition from any input triangles boundary representation (B-rep), by utilizing a very fast computation of the surface inertia. A solid model is so generated at progressive levels of detail. Thi

Parallel Solid Modeling . . .

We introduce a parallel approach to geometric modeling of complex objects and scenes, combining a dataflow streaming of BSP trees with a partition of the object space into independent portions, to be evaluated in parallel with minimal interprocess communication. Binary Space Partition (BSP) is a space index used in graphics for hidden-surface removal and animation. We use BSP trees with fuzzy leaves as a progressive representation of solid meshes. Our approach is implemented as a dataflow with processes that progress concurrently, where each refinement of the input to a process is mapped instantly to a refinement of the output, so that the result is also a stream of progressive refinements. Thi

L'applicazione delle misure agroambientali.Fattori socioeconomici e scelte imprenditoriali

The article explores the socio-economic determinants of the willingness of Veneto region\u2019s farmers to participate in the agri-environmental policies. for the conservation of meadows and pastures in hills and mountains. The quantitative analysis uses the information of the European farm accountancy data network (FADN) in 2003 and it has been realized with a logit regression. It shows the importance of location, smaller total areas, smaller ages and higher average price of land and total amount of payments for adoption. A qualitative analysis was also carried out through face to face interview, in order to evaluate the effects on adoption of some social characteristics of farmers and transaction costs. This analysis demonstrates that farmers have a small perception of environmental problems, agricultural impacts on the environment and AEP goals. Moreover, it has highlighted a generally low environmental attitude among farmers