@David Goodmanson and @dpb and @Steven Lord have all said the polynomial segments are intended to be LOCAL with respect to the break points. I'm adding this as an answer only because I want to explain why it is done that way. Supose I gave you two curves to interpolate using a spline, thus f(x1,y) and f(x2,y).
The two curves should, in theory, be virtually identical. All I have done is horizontally translate the curve in x by some very large number.
Remember, these will be cubic splines. If the cubic segments for these splines were not local things, then in spl2, you would be forming a cubic polynomial where you are raising numbers on the order of 1e8 to the third power, and then adding and subtracting those values to other nubmers. Those polynomials would be unmanagable. You would expect to see massive numerical problems.
You don't want that. Even smaller shifts would cause serious problems. The solution is trivial. Just formulate each segment to be defined as a cubic polynomial, but where the LEFT end of each segment ALWAYS starts at zero, and has length given by diff(breaks). This resolves the numerical problems as much as possible.
