Over the last few days I have been getting interested in basis-splines (or B-Splines). B-Splines are functions $Q(u)$ that are piecewise polynomial of degree $n$, which means that they are made up of sections of polynomials of degree $n$ attached together, that approximate a polygon (called a control polygon).

The sections of polynomials are fitted together smoothly – how smoothly depends on the degree of the polynomials that make up the spline. For example a spline of degree $3$ is made up of sections of cubic polynomials such that the first and second derivatives of the spline $Q(u)$ are continuous at the attachment point but will usually fail to be three times differentiable at these points. More generally – for a spline $Q(u)$ of degree $n$, then $Q(u)$ will be $n-1$ times differentiable but will usually fail to be $n$ times differentiable.

For example, in the following diagram the control polygon is shown in red and the spline, shown in blue, is piecewise cubic

Cubic B-Spline

Cubic B-Spline

The attachment points for this spline are at the integer values along the $x$ axis –  I chose integer points because it makes the algebra considerably easier but in theory there is nothing restricting me to these points. Strictly speaking this is a uniform B-Spline which means that the attachment points are at regular intervals; if the attachment points occur on intervals of different lengths then it is a non-uniform spline which are also very commonly used.

The following graph is a graph of the first derivative of the above spline

Graph of first derivative of Cubic B-Spline

Graph of the first derivative of a cubic B-Spline

 

Clearly this curve is continuous. I have fixed the spline so that the attachment points occur at integer values on the $x$-axis and importantly this curve is continuous at each of these points. We are not necessarily interested in what happens in between these points so it is a bit of a bonus that the derivative is continuous at all intermediate points. Now let’s look at the second derivative

Second Derivative of Cubic B-Spline

Graph of the second derivative of a cubic B-Spline

Again the graph is continuous but we can see just by looking at the graph that it fails to be differentiable at the attachment points – but this is what we expect to happen. The only way that this spline could be more than twice differentiable would be if it was only made up of a single section – but this wouldn’t be a very interesting spline really…

Splines and spline interpolation are very useful in computer-aided design and particularly in computer games design. There is so much that I have learned and have yet to learn about splines that I can’t possible fit everything into a single post so I’m sure that I will be coming back to this topic again very soon.

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