This work describes an automatic algorithm for unstructured mesh regeneration on arbitrarily shaped three-dimensional surfaces. The arbitrary surface may be: a triangulated mesh, a set of points, or an analytical surface (such as a collection of NURBS patches). To be generic, the algorithm works directly in Cartesian coordinates, as opposed to generating the mesh in parametric space, which might not be available in all the cases. In addition, the algorithm requires the implementation of three generic functions that abstractly represent the supporting surface. The first, given a point location, returns the desired characteristic size of a triangular element at this position. The second method, given the current edge in the boundary-contraction algorithm, locates the ideal apex point that forms a triangle with this edge. And the third method, given a point in space and a projection direction, returns the closest point on the geometrical supporting surface. This work also describes the implementation of these three methods to re-mesh an existing triangulated mesh that might present regions of high curvature. In this implementation, the only information about the surface geometry is a set of triangles. In order to test the efficiency of the proposed algorithm of surface mesh generation and implementation of the three abstract methods, results of performance and quality of generated triangular element examples are presented.