Catmull-Rom splines interpolate a set of control points, and are guaranteed to pass through the control points (except for the first and last points).



A Catmull-Rom spline has the following properties:

• It passes through the middle two points, and interpolates the space between them

• The tangents at p1 and p2 are:

Since there are four constraints, we will have four unknowns in the spline equation (i.e., we will have a cubic spline ).

Cubic spline equation:

The goal is to derive a, b, c and d using the constraints.  This will then give the precise form of x(t).



Four linear equations with four unknowns:



This is easily solved using algebra to determine a, b, c, and d!

The four equations can be solved, giving the result:




Given four control points (p0, p1, p2, p3), we can compute a, b, c, and d.  Then, we can sample the curve 

at regular intervals along the interval t=[0, 1] to generate points along the spline spanning the space between p1 and p2.

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