Random reflections in a high dimensional tube

TL;DR

This paper studies the behavior of light rays reflecting inside high-dimensional semi-infinite tubes made of Lambertian material, providing new insights into their exit distributions and extending previous lower-dimensional results.

Contribution

It generalizes previous work on light reflections to higher dimensions ($n \, \geq 3$), deriving new results on exit distributions in these settings.

Findings

Derived the distribution of exit points for high-dimensional tubes

Extended previous 2D and 3D reflection theorems to higher dimensions

Provided mathematical models for Lambertian reflection in $n$-dimensional spaces

Abstract

We consider light ray reflections in $n$-dimensional semi-infinite tube, for $n\geq 3$, made of Lambertian material. The source of light is placed far away from the exit, and the light ray is assumed to reflect so that the distribution of the direction of the reflected light ray has the density proportional to the cosine of the angle with the normal vector. We present new results on the exit distribution from the tube, and generalizations of some theorems from an earlier article, where the dimension was limited to $n=2$ and $3$.