Abstract: Smoothing surface curves are an important step for surface processing applications, such as segmentation, editing and cutting. Various applications require smooth curves that follow the given initial curves closely. One example is surgical planning, in which virtual models are cut open, as in resection planning for liver surgery. There are several approaches to smoothing a surface curve that are based on energy minimization or on interpolating the control points with (piecewise) polynomial curves. These methods, however, do not ensure that the smoothed curve remains close to its initial location. This paper presents a new method for smoothing piecewise linear curves on triangular surface meshes based on a local reduction of the geodesic curvature. The method preserves the closeness to the initial curve. Moreover, the user can adjust the degree of closeness such that the smoothed curve will result in a locally straightest geodesic. We prove that the curve's geodesic curvature decreases in each iteration step, and we use it as an abort criterion. Experiments also confirm robustness to geometric and parametric noise. Finally, we evaluate our method for two surgical planning instances, the decomposition of cerebral aneurysms and resection planning for liver surgery.