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.

翻译：我们致力于生成具有任意粗糙度的表面，并在其表面形成图案。采用两种方法构建粗糙表面。第一种方法利用具有随机频率和传播角度的波函数叠加，通过解析参数方程生成周期性粗糙表面。此类表面的振幅是拉普拉斯-贝尔特拉米算子特征值分析及图案形成生成过程中的重要变量。数值实验表明，随着粗糙表面振幅和频率的增加，图案会变得不规则。为便于推广至闭流形，我们提出第二种粗糙表面构建方法，该方法采用随机节点值与离散化热滤波器。数值结果证明，这两种表面构建方法产生的图案与自然界中观察到的真实动物体表图案具有可比性。