The Geometry of Light Paths for Equiangular Spirals

TL;DR

This paper explores the geometric properties of light paths within equiangular spirals, analyzing reflections, propagation, and critical curves using geometric calculus, revealing insights into light behavior and escape conditions.

Contribution

Introduces geometric calculus for equiangular spirals and investigates light reflections, propagation, and critical curves within this geometric framework.

Findings

Identification of a critical inner curve delimiting right incident light propagation

Analysis of reflection properties at equiangular spirals

Description of light escape conditions within the spiral

Abstract

First geometric calculus alongside its description of equiangular spirals, reflections and rotations is introduced briefly. Then single and double reflections at such a spiral are investigated. It proves suitable to distinguish incidence from the \textit{right} and \textit{left} relative to the radial direction. The properties of geometric light propagation inside the equiangular spiral are discussed, as well as escape conditions and characteristics. Finally the dependence of right and left incidence from the source locations are examined, revealing a well defined inner \textit{critical} curve, which delimits the area of purely right incident propagation. This critical curve is self similar to the original equiangular spiral.