Computational and qualitative aspects of evolution of curves driven by curvature and external force

We propose a direct method for solving the evolution of plane curves satisfying the geometric equation v= β(x, k, ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ⊂ R2 at a point x ∈ Γ . We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves. The governing equations include a nontrivial tangential velocity functional yielding uniform redistribution of grid points along the evolving family of curves preventing thus numerically computed solutions from forming various instabilities. We also propose a full space-time discretization of the governing system of equations and study its experimental order of convergence. Several computational examples of evolution of plane curves driven by curvature and external force as well as the geodesic curvatures driven evolution of curves on various complex surfaces are presented in this paper.