Computing the shortest path between two points or a set of points on a point cloud representing a manifold is a crucial problem in Computational Science, particularly in Computer Graphics and CAD/CAM. Most methods calculate geodesic paths on a polygonal model of the manifold, while only a few compute them directly on the point cloud. When the point cloud has noise, constructing a triangular mesh is challenging and results in inaccurate geodesic path calculations. This work calculates the geodesic path by minimizing the discrete geodesic curvature of the connecting curve, using an iterative Newton minimization and a projection procedure based on directed point-projection and Gabriel neighborhoods. The proposed algorithm has quadratic convergence and produces accurate results in synthetic and free-form point clouds, outperforming triangle mesh-based methods in noisy point clouds. The paper concludes with a demonstration of geodesic path computation in diverse free-form point-clouds and a case study in template-based digital surface reconstruction. The source code of the proposed method can be found in https://github.com/agalex1974/LibGeodesicOPPGC .