The task of filling holes in three-dimensional triangular meshes is critical to a wide variety of scanning and printing operations. We will discuss some recent solutions to the hole-filling problem and discuss their strengths and weaknesses. In particular, a high-quality filling typically comes at the cost of high computation time and memory usage. We propose a new algorithm for triangulation (a necessary step in many existing hole-filling techniques) that allows for a smooth, user-defined trade-off between the quality of the filling and the computational resources required to compute it. We also offer empirical evidence as to the utility of the algorithm.