A [[Closure|closed]] [[Spline|spline]] $P(t)$ is a form of [[Parametric Spline]] that forms a closed loop / is [[Periodic Function|periodic]] in terms of $t$. For a Cubic Spline, the [[System of Linear Equations]] would be this: $ \mat{ 4 &1 & 0 & \cdots & 0 & 1 \\ 1 & 4 & 1 & \cdots & 0 &0 \\ 0 & 1 & 4 & 1 & \cdots & 0 \\ & & \ddots & \ddots & \ddots & 1 \\ 1 & & & &1 & 4 } \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} } $ This is similar to the one for a simple Parametric Spline except with an added '1' in the top right and top left of the first [[Matrix]].