Multiquadric interpolation: Surface fitting in three-dimensional space

Using mathematics to solve a problem does not always yield a perfect or absolute answer but may instead yield an approximate solution. We can try to approximate the solution as precisely as possible by using the mathematical tools and skills that are available to us or we could try to discover new methods which would enable us to find good approximations. It is important that we have precise approximating tools to begin with, so that we may preserve as much accuracy as possible. We can find such problems in the world around us. For instance, if we try to construct a topographical map of a mountainous region, first we gather data by measuring some elevations and locations. The data is then used to construct the map. We now realize that because measuring every dip and valley of the area would be an impossible task, the map must be constructed from a set of random points. The next step is either to guess about the elevations between the data points, if there are enough points close enough together, or to estimate these elevations mathematically. Since we would like to finish constructing the map by taking small regions around the known data points and finding approximating functions which, when graphed, will represent as precisely as possible the surface of the region, an entirely new problem arises. These surfaces around the known points cannot be easily calculated by use of simple functions. We now need to use these few, random data point to find an approximating surface by means of an interpolation method