4.10 TENSION SPLINES Splines are a wonderful tool for approximation, but they can still exhibit some poor behavior. Consider the data set plotted in Fig. 4.25. Obviously, this represents a function with a severe jump near x — 0.5, but there is no sign of oscillatory behavior. However, a B-spline representation of this data (Fig. 4.26) shows small wiggles on either side of a sharp front. This is fundamentally an artifact of the steep gradient in the data, but in other contexts a spline fit can display behavior that does not match the "sense" of the data. One way to avoid the problem is the notion of a taut spline or tension spline, an idea that appears to have been first published by Schweikert [17], but which also owes a lot to the work of A. K. Cline [4]; we relied heavily on a short paper of Marusic and Rogina [12] in our presentation here. Imagine that the curve in Fig 4.26 is a piece of string that is constrained to pass through small loops at the data points. If we were...
4.10 TENSION SPLINES Splines are a wonderful tool for approximation, but they can still exhibit some poor behavior. Consider the data set plotted in Fig. 4.25. Obviously, this represents a function with a severe jump near x — 0.5, but there is no sign of oscillatory behavior. However, a B-spline representation of this data (Fig. 4.26) shows small wiggles on either side of a sharp front. This is fundamentally an artifact of the steep gradient in the data, but in other contexts a spline fit can display behavior that does not match the "sense" of the data. One way to avoid the problem is the notion of a taut spline or tension spline, an idea that appears to have been first published by Schweikert [17], but which also owes a lot to the work of A. K. Cline [4]; we relied heavily on a short paper of Marusic and Rogina [12] in our presentation here. Imagine that the curve in Fig 4.26 is a piece of string that is constrained to pass through small loops at the data points. If we were...