We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bézier curves. In the second approach, we consider Bézier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bézier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.