Bezier end-conditions

For approximating schemes we can design end-conditions by analogy with the B-splines, where the most widely used end-condition is that called Bezier end-conditions. 

In fact all three approaches are equivalent, and we shall illustrate the first with Bezier end-conditions for box-splines, and the second with Lagrange conditions for the four-point scheme. 

The end-conditions described above cover two distinct cases, those of interpolating schemes, which are likened to Lagrange interpolation, and those of B-splines, likened to the Bezier end-conditions. The schemes which interpolate when the data lies on a cubic or higher polynomial do not really fit either of these cases. They are almost interpolating (when the data is really smooth) but not quite. Somebody needs to play with these schemes to find out how they currently misbehave at the ends and what kinds of control are required to make them do what the curve designer wants. 

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