An introduction to the mathematics and construction of splines

Everyone that has ever tried to apply simple linear interpolation to find a value between pairs of data points will be only too aware that such attempts are extremely unlikely to provide reliable results if the data being used is anything other than broadly linear. In an attempt to deal with inherent non-linearity, the next step usually involves some sort of polynomial interpolation. This generally leads to far more stable and robust interpolation and fitting, but is also potentially a difficult area as the end points, monotonicity, convexity and continuity of derivatives all make their influences felt in often contradictory ways. One of the most popular ways of dealing with these issues is to use splines. In their most general form, splines can be considered as a mathematical model that associate a continuous representation of a curve or surface with a discrete set of points in a given space. Spline fitting is an extremely popular form