Spline¶

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A Spline is a closed polynomial curve of a curve parameter $t$ interpolating the given sequential list of Points. Its Degree controls the polynomial order of the curve with higher degrees leading to smoother curves. The mapping

$\begin{eqnarray*} t \mapsto \gamma(t)=(X(t),Y(t)),\quad t \in [0,1] \end{eqnarray*}$

is chosen such that $X(t),Y(t)$ are splines of the desired order and the curve parameterization in $t$ follows the linearized arc length. This means the interpolation points $t_i$ where $\gamma(t_i)=P_i$ for the i-th point of the Points list are approximately the ratio of the length of the polygon up to $P_i$ to the total length. For the best interpolation result the linear sampling provided by the Points should already yield a good approximation to the desired curve.