A [[Spline]] is a type of [[Piecewise]] [[Polynomial]] [[Curve]] used to describe a [[Hermite Interpolation]] between multiple points. Typically the restrictions imposed on a Spline $S(x)$ is that it is desired for it to be [[Continuous]] up to the 2nd derivative $S(x)\in \mathcal C^{2}[a,b]$. >[!info] Figure ![[Pasted image 20260209150548.png|invert_Sepia]] ### Constraints If you want to interpolate between $n$-points, $S(t)$ will be composed of $n$ different splines. To keep $S(t)\in \mathcal C^{0}$ would be $2n$ constraints (each end of start of contiguous splines $S_i$ must connect). To force $S(t)\in \mathcal C^{1}$ would require $2n+(n-1)$ constraints. For $S(t)\in\mathcal C^{2}$, $2n+(n-1)+(n-1)$, or $4n-2$. $\large \# \text{Constraints}\set{S(t)\in \mathcal C^{m}[a,b]} = 2n+ \op{max}(0,(m-1)(n-1)) $ For a 'natural spline', we also required that the second derivative $S''$ at the two end points are $0$.