  We propose an extension of the Allen-Cahn model for pattern synthesis on two
dimensional curved surfaces.
  This model is based on a single PDE and it offers improved ability of
controlling the type of generated surface patterns via the chosen
reaction-diffusion coefficient, thus, obtaining patterns in form of spots,
inverted spots, or stripes. We investigate the dependence of the type of the
obtained pattern on the new proposed reaction term. An efficient operator
splitting scheme is used to discretize the model on a surface. Experiments on
surfaces with varying initial conditions illustrate a variety of patterns.
