183,682 research outputs found
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
Radon Transform in Finite Dimensional Hilbert Space
Novel analysis of finite dimensional Hilbert space is outlined. The approach
bypasses general, inherent, difficulties present in handling angular variables
in finite dimensional problems: The finite dimensional, d, Hilbert space
operators are underpinned with finite geometry which provide intuitive
perspective to the physical operators. The analysis emphasizes a central role
for projectors of mutual unbiased bases (MUB) states, extending thereby their
use in finite dimensional quantum mechanics studies. Interrelation among the
Hilbert space operators revealed via their (finite) dual affine plane geometry
(DAPG) underpinning are displayed and utilized in formulating the finite
dimensional ubiquitous Radon transformation and its inverse illustrating phase
space-like physics encoded in lines and points of the geometry. The finite
geometry required for our study is outlined.Comment: 8page
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