B-Spline vs \(\pi\)
How B-splines \(\beta\) lead to the mathematical constant \(\pi.\)
Value of a B-Spline at the Origin
Consider the B-spline \(\beta^{n}\) of degree \(n\) and let us focus on the value \(\beta^{n}(0)\) that this function takes at the origin. Then, it turns out that
\[\pi=6\,\lim_{n\rightarrow\infty}\frac{1}{n+1}\,\left(\beta^{n}(0)\right)^{-2}.\]
In the following Jupyter Lab notebook, we plot a visual representation of that fact.
Jupyter Lab notebook
Alternating Sum of Samples
Another relation between B-splines and \(\pi\) arises if one computes the sum of all integer samples of B-splines, with an alternating sign. Then, it turns out that
\[\pi=2\,\lim_{n\rightarrow\infty}\left(\frac{1}{2}\,\sum_{k\in{\mathbb{Z}}}\,\left(-1\right)^{k}\,\beta^{n}(k)\right)^{-\frac{1}{n+1}}.\]
In the following Jupyter Lab notebook, we plot a visual representation of that fact.
Jupyter Lab notebook