Realistic pattern formations on surfaces by adding arbitrary roughness

TL;DR

This paper introduces two methods for generating surfaces with arbitrary roughness to produce natural pattern formations, demonstrating their effectiveness through numerical experiments and comparisons to real animal patterns.

Contribution

It presents novel methods for constructing rough surfaces using wave superpositions and heat filters, enabling realistic pattern formation analysis.

Findings

Patterns become more irregular with increased amplitude and frequency.

Both methods produce patterns comparable to real animal surfaces.

Numerical experiments validate the effectiveness of the proposed methods.

Abstract

We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface {construction methods} yield comparable patterns to those {observed} in real-life animals.