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We are interested in generating surfaces with arbitrary roughness and formingpatterns on the surfaces. Two methods are applied to construct rough surfaces.In the first method, some superposition of wave functions with randomfrequencies and angles of propagation are used to get periodic rough surfaceswith analytic parametric equations. The amplitude of such surfaces is also animportant variable in the provided eigenvalue analysis for the Laplace-Beltramioperator and in the generation of pattern formation. Numerical experiments showthat the patterns become irregular as the amplitude and frequency of the roughsurface increase. For the sake of easy generalization to closed manifolds, wepropose a second construction method for rough surfaces, which uses randomnodal values and discretized heat filters. We provide numerical evidence thatboth surface {construction methods} yield comparable patterns to those{observed} in real-life animals.

Realistic pattern formations on surfaces by adding arbitrary roughness