Its degree . Its breakpoints . Its level of derivative continuity at the breakpoints. Extensions of this approach can be used to extract the B-spline coefficients of rational spline curves as well as polynomial and rational spline surfaces. In the rational case, the evaluator that needs to be supplied must allow you to get at the numerator and denominator of the expression separately.
Similar considerations hold when working with rational surfaces.
This typically requires that you must have access to homogeneous evaluators for the defining curves. My colleague, Michael Kallay realized this and replaced this approach with one that found the B-spline coefficients by setting up and solving a system of linear equations. Because of the way that we had designed the software, it was possible to unplug the deBoor-Fix machine and plug in the linear algebra machine very quickly with very little disruption to the overall scheme of things. I hope that I have conveyed something of the flavor of this effort. It was a nice combination of technical challenges, teamwork, and the unreasonable effectiveness of mathematics.