Physics from Angular Projection of Rectangular Grids

TL;DR

This paper develops a mathematical model for the angular projection of rectangular grids, revealing non-distinctness angles and offering a method to determine unknown grid geometry from projections, with applications in physics and data analysis.

Contribution

It introduces a comprehensive mathematical framework for analyzing the angular projection of rectangular grids, including non-distinctness angles and a technique to infer grid geometry from projections.

Findings

Identified angles with non-distinct projections.

Derived formulas for the number of distinct projected points.

Demonstrated application in determining unknown grid geometry.

Abstract

In this paper, we present a mathematical model for the angular projection of a rectangular arrangement of points in a grid. This simple, yet interesting problem, has both a scholarly value and applications for data extraction techniques to study the physics of various systems. Our work can interest undergraduate students to understand subtle points in the angular projection of a grid and describes various quantities of interest in the projection with completeness and sufficient rigour. We show that for certain angular ranges, the projection has non-distinctness, and calculate the details of such angles, and correspondingly, the number of distinct points and the total projected length. We focus on interesting trends obtained for the projected length of the grid elements and present a simple application of the model to determine the geometry of an unknown grid whose spatial extensions are known, using measurement of the grid projection at two angles only. Towards the end, our model is shown to have potential applications in various branches of physical sciences including crystallography, astrophysics and bulk properties of materials.