We will consider two aspects concerning the reconstruction of surfaces from unstructured distributed data. The first is detecting the discontinuities the surface may have in order to reproduce them as accurately as possible. Finding the discontinuity curves, usually called faults (or gradient faults when gradient discontinuities are considered), is actually an important and non-trivial issue in itself, with several applications, for example in image processing and geophysics.

The second aspect is obtaining the continuous model describing the surface from the set of scattered points: the lack of structure in the data requires approximation methods which automatically adapt to the distribution and shape of the data themselves. We will discuss an adaptive approach to this issue based on spline spaces.