Catmull-Rom splines

The Catmull-Rom splines is a method that approximate a set of points (named control points) with a smooth polynomial function that is piecewise-defined. One of the properties of the Catmull-Rom spline is that the curve will pass through all of the control points.

Equations

Two points on each side of the desired portion are required. In other words, points $ P_{i-1} $ and $ P_{i+2} $ are needed to calculate the spline between points $P_i$ and $P_{i+1}$.

Given points $P_{i-1}$, $P_{i}$, $P_{i+1}$ and $P_{i+2}$, the coordinates of a point $P$ located between $P_i$ and $P_{i+1}$ are calculated as:

$$ P = \frac{1}{2} . \left[ \begin{matrix} 1 & t & t^2 & t^3 \end{matrix} \right]. \left[ \begin{matrix} 0 & 2 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 2 & -5 & 4 & -1 \\ -1 & 3 & -3 & 1 \end{matrix} \right]. \left[ \begin{matrix} P_{i-1} & P_{i} & P_{i+1} & P_{i+2} \end{matrix} \right]^\top $$

General case: tension

The previous equation is a particular case of the general geometry matrix given by the following equation:

$$ P = \left[ \begin{matrix} 1 & t & t^2 & t^3 \end{matrix} \right]. \left[ \begin{matrix} 0 & 1 & 0 & 0 \\ -\tau & 0 & \tau & 0 \\ 2.\tau & \tau-3 & 3-2\tau & -\tau \\ -\tau & 2-\tau & \tau-2 & \tau \end{matrix} \right]. \left[ \begin{matrix} P_{i-1} & P_{i} & P_{i+1} & P_{i+2} \end{matrix} \right]^\top $$

The parameter $\tau$ modify the tension of the curve. The following figure illustrates the influence of the parameter $\tau$ on the curve. Note that $\tau=\frac{1}{2}$ is commonly used (as in the particular case presented previously).

Derivative

As the spline is $C^1$ continuous it is possible to compute the derivative for any value of $t$. Moreover, as the definition of the spline is a polynomial, it is quite trivial to compute the derivative at a given point:

$$ P = \left[ \begin{matrix} 0 & 1 & 2.t & 3.t^2 \end{matrix} \right]. \left[ \begin{matrix} 0 & 1 & 0 & 0 \\ -\tau & 0 & \tau & 0 \\ 2.\tau & \tau-3 & 3-2\tau & -\tau \\ -\tau & 2-\tau & \tau-2 & \tau \end{matrix} \right]. \left[ \begin{matrix} P_{i-1} & P_{i} & P_{i+1} & P_{i+2} \end{matrix} \right]^\top $$