In a previous post I spoke a little bit about B-Splines and in particular, Uniform B-Splines. I didn’t really go into much detail about how they are defined, how we decide what our b-spline is going to look like or even what an explicit expression for a spline would look like given our set of control points.

I decided to type up a LaTeX document which introduces the theory of b-splines. The document looks at b-splines from a practical perspective and so it doesn’t get too bogged down with the analysis side of things but concentrates on the tools required to find explicit expressions for b-splines.

Download the full document onĀ Uniform B-Splines here.

The document contains expressions for piecing together a b-spline. I spent a good number of hours deriving these expressions and they got very messy to deal with but, eventually, persistence prevailed. It is likely that someone, somewhere has already derived these formulae but by doing the work myself I was able to learn so much about these splines that I wouldn’t have appreciated if I had just read someone else’s work.

This is one of the most important things about becoming better as a mathematician – being prepared to spend time with a problem and being willing to make a lot of mistakes until you get things right. Persistence is rarely a bad thing. I make no apology for the length of some of the expressions – I have not made any real attempt to simplify the expressions as I feel that it would be a glorious waste of time to try and do so. If anyone out there would like to have a go at tidying up the expressions then feel free to go right ahead.