Snell's law for particles moving on piecewise homogeneous two dimensional surface with linear boundaries

TL;DR

This paper generalizes Snell's law to particles moving on a piecewise homogeneous 2D surface, modeling their trajectories and behaviors near edges, with applications to animal trail patterns.

Contribution

It introduces a generalized Snell's law for particles on piecewise surfaces and predicts natural trail behaviors near edges, extending classical optics principles.

Findings

Particles tend to follow edges when moving between regions.

Three trajectory behaviors near corners: edge following, partial, avoidance.

Model aligns with observed ant trail behaviors.

Abstract

We study the properties of least time trajectories for particles moving on a two dimensional surface which consists of piecewise homogeneous regions. The particles are assumed to move with different constant speeds on different regions and on the boundary between regions. The speed of the particle is assumed to be highest when it moves along the edges formed by the boundary of two regions. We get an analogous behavior to Snell's Law of light refraction, but in a more generalized form. The model could be used for studying properties of animal and insect trails which tend to form predominantly along edges. The model predicts three types of behavior for the trajectories near a corner forming edge: fully edge following, partial edge following and complete avoidance of the edge, which are indeed observed in natural ant trails.