Curves are often used as anatomical features to match surfaces that represent biological objects, such as the human brain. Automated and semi-automated methods for extracting these curves usually rely on local properties of the surfaces such as the mean surface curvature without considering the global appearance of the curves themselves. These methods may require additional human intervention, and sometimes produce erroneous results. In this paper, we present an algorithm that is based on the fast marching method (FMM) to extract weighted geodesic curves. Instead of directly using the local image properties as a weight function, we use the surface properties, together with the global properties of the curves, to compute a weight function. This weight function is then used by the FMM to extract curves between given points. The general framework can be used to extract curves with different global properties. The resulting curves are guaranteed to be weighted geodesic curves without cusps usually introduced by intermediate points through which the curves are forced to pass. We show some results on both a simulated image and a highly convoluted human brain cortical surface.