![[Pasted image 20260204155318.png|invert_B]] A parametric [[Spline]] $P(x)$ is a [[Spline]] ([[Hermite Interpolation]]) modeled by a [[Parametric Function]] in terms of a single parameter $t\in[0,n]$ such that $P(i) = (x_{i},y_{i})$. These types of splines are useful as they scale to interpolate between points of any [[Dimension]]. The most common form of parametric spline is a cubic spline. $ \mat{ 4 &1 & 0 & \cdots & 0 & 0 \\ 1 & 4 & 1 & \cdots & & \\ 0 & 1 & 4 & 1 & \cdots & \\ & & \ddots & \ddots & \ddots & } \mat{ x_{1}'' & y_{1}'' \\ x_{2}'' & y_{2}'' \\ \vdots & \vdots \\ x_{n-1}'' & y_{n-1}'' } = 6\mat{ \Delta^{2}x_{0} & \Delta^{2}y_{0} \\ \Delta^{2} x_{1} & \Delta^{2}y_{1} \\ \vdots & \vdots \\ \Delta^{2} x_{n-1} & \Delta^{2} y_{n-1} } $ Note that if this is a [[Closed Spline]] this is differant.