This article describes algorithms and data-structures for the fast construction of three-dimensional triangulations from large sets of scattered data-points. The triangulations have a guaranteed error bound, i.e. all the data-points lie within a pre-specified distance from the triangulation. Three different methods for choosing triangulation vertices are presented, based on interpolation, and L₂ and L∞-optimization of the error over subsets of the data-points. The main focus of this article will be on devising a simple and fast algorithm for constructing an approximating triangulation of a very large set of points. We propose the use of adapted dynamic data structures and excessive caching of information to speed up the computation and show how the method can be extended to approximate multiple dependent datasets in higher-dimensional approximation problems.