In this paper we investigate two non-local geometric geodesic curvature driven flows of closed curves preserving either their enclosed surface area or their total length on a given two-dimensional surface. The method is based on projection of evolved curves on a surface to the underlying plane. For such a projected flow we construct the normal velocity and the external nonlocal force. The evolving family of curves is parametrized by a solution to the fully nonlinear parabolic equation for which we derive a flowing finite volume approximation numerical scheme. Finally, we present various computational examples of evolution of the surface area and length preserving flows of surface curves. We furthermore analyse the experimental order of convergence. It turns out that the numerical scheme is of the second order of convergence.
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